Empirical Gramians and FriendsChristian Himpe
2016-10-11
CSC
Outline
1 My Research Interests
2 A Tour Through my Scientific Deeds
3 Lift-Off for MathEnergy
4 Scientific Sideshows
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 242
Research Interests
in a Nutshell
Combined State and Parameter Reduction
for Nonlinear Input-Output Systems
Modeling (Complex) Networks
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 342
Notation
Nonlinear Input-Output System
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Input Function u Rge0 rarr RM
State Trajectory x Rge0 rarr RN
Output Trajectory y Rge0 rarr RQ
Parameter θ isin RP
Vector-Field f Rge0 times RN times RM times RP rarr RN
Output Functional g Rge0 times RN times RM times RP rarr RQ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 442
Combined Reduction
Reduced Order Model
xr (t) = fr (t xr (t) u(t) θr )
yr (t) = gr (t xr (t) u(t) θr )
xr (0) = xr 0
State Trajectory xr Rge0 rarr Rn
Output Trajectory yr Rge0 rarr RQ
Parameter θr isin Rp
Vector-Field fr Rge0 times Rn times RM times Rp rarr Rn
Output Functional gr Rge0 times Rn times RM times Rp rarr RQ
Reduced State Dimension n NReduced Parameter Dimension p P
Accuracy y(θ)minus yr (θr ) 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 542
Projection-Based Reduction
Projection-Based Reduced Order Model
xr (t) = Vf (tUxr (t) u(t)Πθr )
yr (t) = g(tUxr (t) u(t)Πθr )
xr (0) = Vx0
θr = Λθ
Reducing Truncated State Projection V isin RntimesN
Reconstructing Truncated State Projection U isin RNtimesn
Bi-Orthogonality VU = 1
Reducing Truncated Parameter Projection Λ isin RptimesP
Reconstructing Truncated Parameter Projection Π isin RPtimesp
Bi-Orthogonality ΛΠ = 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 642
Previously
Dynamic Causal Modelling
Combined State and Parameter Reduction
Empirical-Cross-Gramian-BasedGreedy-Optimization-Based
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 742
Dynamic Causal Modelling (DCM)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 842
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Outline
1 My Research Interests
2 A Tour Through my Scientific Deeds
3 Lift-Off for MathEnergy
4 Scientific Sideshows
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 242
Research Interests
in a Nutshell
Combined State and Parameter Reduction
for Nonlinear Input-Output Systems
Modeling (Complex) Networks
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 342
Notation
Nonlinear Input-Output System
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Input Function u Rge0 rarr RM
State Trajectory x Rge0 rarr RN
Output Trajectory y Rge0 rarr RQ
Parameter θ isin RP
Vector-Field f Rge0 times RN times RM times RP rarr RN
Output Functional g Rge0 times RN times RM times RP rarr RQ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 442
Combined Reduction
Reduced Order Model
xr (t) = fr (t xr (t) u(t) θr )
yr (t) = gr (t xr (t) u(t) θr )
xr (0) = xr 0
State Trajectory xr Rge0 rarr Rn
Output Trajectory yr Rge0 rarr RQ
Parameter θr isin Rp
Vector-Field fr Rge0 times Rn times RM times Rp rarr Rn
Output Functional gr Rge0 times Rn times RM times Rp rarr RQ
Reduced State Dimension n NReduced Parameter Dimension p P
Accuracy y(θ)minus yr (θr ) 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 542
Projection-Based Reduction
Projection-Based Reduced Order Model
xr (t) = Vf (tUxr (t) u(t)Πθr )
yr (t) = g(tUxr (t) u(t)Πθr )
xr (0) = Vx0
θr = Λθ
Reducing Truncated State Projection V isin RntimesN
Reconstructing Truncated State Projection U isin RNtimesn
Bi-Orthogonality VU = 1
Reducing Truncated Parameter Projection Λ isin RptimesP
Reconstructing Truncated Parameter Projection Π isin RPtimesp
Bi-Orthogonality ΛΠ = 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 642
Previously
Dynamic Causal Modelling
Combined State and Parameter Reduction
Empirical-Cross-Gramian-BasedGreedy-Optimization-Based
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 742
Dynamic Causal Modelling (DCM)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 842
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Research Interests
in a Nutshell
Combined State and Parameter Reduction
for Nonlinear Input-Output Systems
Modeling (Complex) Networks
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 342
Notation
Nonlinear Input-Output System
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Input Function u Rge0 rarr RM
State Trajectory x Rge0 rarr RN
Output Trajectory y Rge0 rarr RQ
Parameter θ isin RP
Vector-Field f Rge0 times RN times RM times RP rarr RN
Output Functional g Rge0 times RN times RM times RP rarr RQ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 442
Combined Reduction
Reduced Order Model
xr (t) = fr (t xr (t) u(t) θr )
yr (t) = gr (t xr (t) u(t) θr )
xr (0) = xr 0
State Trajectory xr Rge0 rarr Rn
Output Trajectory yr Rge0 rarr RQ
Parameter θr isin Rp
Vector-Field fr Rge0 times Rn times RM times Rp rarr Rn
Output Functional gr Rge0 times Rn times RM times Rp rarr RQ
Reduced State Dimension n NReduced Parameter Dimension p P
Accuracy y(θ)minus yr (θr ) 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 542
Projection-Based Reduction
Projection-Based Reduced Order Model
xr (t) = Vf (tUxr (t) u(t)Πθr )
yr (t) = g(tUxr (t) u(t)Πθr )
xr (0) = Vx0
θr = Λθ
Reducing Truncated State Projection V isin RntimesN
Reconstructing Truncated State Projection U isin RNtimesn
Bi-Orthogonality VU = 1
Reducing Truncated Parameter Projection Λ isin RptimesP
Reconstructing Truncated Parameter Projection Π isin RPtimesp
Bi-Orthogonality ΛΠ = 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 642
Previously
Dynamic Causal Modelling
Combined State and Parameter Reduction
Empirical-Cross-Gramian-BasedGreedy-Optimization-Based
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 742
Dynamic Causal Modelling (DCM)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 842
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Notation
Nonlinear Input-Output System
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Input Function u Rge0 rarr RM
State Trajectory x Rge0 rarr RN
Output Trajectory y Rge0 rarr RQ
Parameter θ isin RP
Vector-Field f Rge0 times RN times RM times RP rarr RN
Output Functional g Rge0 times RN times RM times RP rarr RQ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 442
Combined Reduction
Reduced Order Model
xr (t) = fr (t xr (t) u(t) θr )
yr (t) = gr (t xr (t) u(t) θr )
xr (0) = xr 0
State Trajectory xr Rge0 rarr Rn
Output Trajectory yr Rge0 rarr RQ
Parameter θr isin Rp
Vector-Field fr Rge0 times Rn times RM times Rp rarr Rn
Output Functional gr Rge0 times Rn times RM times Rp rarr RQ
Reduced State Dimension n NReduced Parameter Dimension p P
Accuracy y(θ)minus yr (θr ) 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 542
Projection-Based Reduction
Projection-Based Reduced Order Model
xr (t) = Vf (tUxr (t) u(t)Πθr )
yr (t) = g(tUxr (t) u(t)Πθr )
xr (0) = Vx0
θr = Λθ
Reducing Truncated State Projection V isin RntimesN
Reconstructing Truncated State Projection U isin RNtimesn
Bi-Orthogonality VU = 1
Reducing Truncated Parameter Projection Λ isin RptimesP
Reconstructing Truncated Parameter Projection Π isin RPtimesp
Bi-Orthogonality ΛΠ = 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 642
Previously
Dynamic Causal Modelling
Combined State and Parameter Reduction
Empirical-Cross-Gramian-BasedGreedy-Optimization-Based
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 742
Dynamic Causal Modelling (DCM)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 842
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Combined Reduction
Reduced Order Model
xr (t) = fr (t xr (t) u(t) θr )
yr (t) = gr (t xr (t) u(t) θr )
xr (0) = xr 0
State Trajectory xr Rge0 rarr Rn
Output Trajectory yr Rge0 rarr RQ
Parameter θr isin Rp
Vector-Field fr Rge0 times Rn times RM times Rp rarr Rn
Output Functional gr Rge0 times Rn times RM times Rp rarr RQ
Reduced State Dimension n NReduced Parameter Dimension p P
Accuracy y(θ)minus yr (θr ) 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 542
Projection-Based Reduction
Projection-Based Reduced Order Model
xr (t) = Vf (tUxr (t) u(t)Πθr )
yr (t) = g(tUxr (t) u(t)Πθr )
xr (0) = Vx0
θr = Λθ
Reducing Truncated State Projection V isin RntimesN
Reconstructing Truncated State Projection U isin RNtimesn
Bi-Orthogonality VU = 1
Reducing Truncated Parameter Projection Λ isin RptimesP
Reconstructing Truncated Parameter Projection Π isin RPtimesp
Bi-Orthogonality ΛΠ = 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 642
Previously
Dynamic Causal Modelling
Combined State and Parameter Reduction
Empirical-Cross-Gramian-BasedGreedy-Optimization-Based
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 742
Dynamic Causal Modelling (DCM)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 842
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Projection-Based Reduction
Projection-Based Reduced Order Model
xr (t) = Vf (tUxr (t) u(t)Πθr )
yr (t) = g(tUxr (t) u(t)Πθr )
xr (0) = Vx0
θr = Λθ
Reducing Truncated State Projection V isin RntimesN
Reconstructing Truncated State Projection U isin RNtimesn
Bi-Orthogonality VU = 1
Reducing Truncated Parameter Projection Λ isin RptimesP
Reconstructing Truncated Parameter Projection Π isin RPtimesp
Bi-Orthogonality ΛΠ = 1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 642
Previously
Dynamic Causal Modelling
Combined State and Parameter Reduction
Empirical-Cross-Gramian-BasedGreedy-Optimization-Based
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 742
Dynamic Causal Modelling (DCM)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 842
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Previously
Dynamic Causal Modelling
Combined State and Parameter Reduction
Empirical-Cross-Gramian-BasedGreedy-Optimization-Based
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 742
Dynamic Causal Modelling (DCM)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 842
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Dynamic Causal Modelling (DCM)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 842
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Dynamic Causal Modelling [Friston et alrsquo03]
AimReconstruction of brain connectivity
from functional neuroimaging data (fMRIfNIRS EEGMEG)
ConceptHierarchical Model
1 Network Submodel (Dynamic Submodel)2 Observation Submodel (Forward Submodel)
Connectivity Parametrization
Effective Connectivity (amp Lateral Connectivity)
SIMO Models (amp MIMO Models)
Bayesian Inference
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 942
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
fMRI amp fNIRS Model
Dynamic Submodel (Taylor)
x(t) = f (x u θ)
asymp f (0 0 θ) +partf
partxx(t) +
partf
partuu(t)
= A(θ)x(t) + Bu(t)
Aij(θ) = θik+j rarr x isin Rkθ isin Rk2
Forward Submodel (Balloon)
si (t) = xi (t) minus κsi (t) minus γ(fi (t) minus 1)
fi (t) = si (t)
vi (t) =1
τ(fi (t) minus vi (t)
1α )
qi (t) =1
τ(
1
ρfi (t)(1 minus ((1 minus ρ))
1fi (t) ) minus vi (t)
1αminus1
qi (t))
yi (t) = V0(k1(1 minus qi (t)) + k2(1 minus vi (t)))
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1042
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
EEG amp MEG Model [Moran et alrsquo07]
Neural Mass Model(v(t)x(t)
)=
(0 1minusT minusT 2
)(v(t)x(t)
)+
(0
A(θ)
)ς(Kx(t)) +
(0B
)u(t)
y(t) = Lx(t)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1142
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Efficient DCM Implemention
Bayesian Inference using
EM AlgorithmE xpectation
WeightedTikhonov regularizedLeast-squaresTemporal Correlations
M aximization
Various additional modifications
See my diploma thesis doi106084m9figshare1027354v1
Now what if one wants to infer on large networks (N gt 6rarr P gt 36)
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Combined State and Parameter Reduction
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1342
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1442
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Cross Gramian [Fernando amp Nicholsonrsquo83]
Linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
Hankel operator
H = OC
Cross Gramian (assume a square system M = Q)
WX = CO =
int infin0
eAt BC eAt dt isin RNtimesN
rArr AWX + WXA = minusBC
Cross Gramianrsquos central property
CAminus1B = (CAminus1B)ᵀ rArr σi (H) = |λi (WX )|
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1542
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Cross-Gramian-Based Reduction
Balancing Transformation (for a symmetric system) [Aldhaherirsquo91]
WXEVD= TΛTminus1
Approximate Balancing Transformation [Sorensen amp Antoulasrsquo02]
WXSVD= UDV
Direct Truncation [Himpe amp Ohlbergerrsquo14]
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1642
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Empirical Cross Gramian [Streif et alrsquo06]
Empirical linear cross Gramian
WX =
int infin0
eAt BC eAt dt =
int infin0
(eAt B
)(eA
ᵀt Cᵀ)ᵀ dt
Empirical cross Gramian
WX =1
KLM
Ksumk=1
Lsuml=1
Msumm=1
1
ckdl
int infin0
Ψklm(t) dt isin RNtimesN
Ψklmij (t) = 〈xkmi (t)minus xi y
ljm(t)minus ym〉
State trajectory xkm(t) for the impulse input u(t) = ckemδ(t)
Output trajectory y lj(t) for the initial state x0 = dlej
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1742
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Joint Gramian [Himpe amp Ohlbergerrsquo14]
Augmented system (treat parameters as constant states)(x(t)
θ(t)
)=
(f (t x(t) u(t) θ(t))
0
)y(t) = g(t x(t) u(t) θ(t))(
x(0)θ(0)
)=
(x0
θ0
)
Joint Gramian (empirical cross Gramian of the augmented system)
WJ =
(WX Wm
0 0
)
Lower blocks are zero as parameter-states are uncontrollable
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1842
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Cross-Identifiability Gramian
Cross-Identifiability Gramian (Schur complement of symmetric part of WJ)
WI = minus1
2W ᵀ
m(WX + W ᵀX )minus1Wm
WI encodes the observability of parameters
Schur-complement can be approximated efficiently
Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 1942
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Cross-Gramian-Based Combined Reduction
1 Compute Empirical Joint Gramian WJ
2 State-Space Direct Truncation
WXSVD= UDV rarr U =
(U1 U2
)rarr V1 = Uᵀ
1
3 Parameter-Space Direct Truncation
WISVD= Π∆Λrarr Π =
(Π1 Π2
)rarr Λ1 = Πᵀ
1
Empirical-Cross-Gramian-Based Reduced Order Model
xr (t) = V1f (tU1xr (t) u(t)Π1θr )
yr (t) = g(tU1xr (t) u(t)Π1θr )
xr (0) = V1x0
θr = Λ1θ
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2042
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Non-Symmetric Cross Gramian [Himpe amp Ohlbergerrsquo16]
A cross Gramian for non-square and non-symmetric systems
Cross Gramian Superposition (B =(b1 bM
) C =
(c1 cQ
)ᵀ)
WX =M=Qsumi=1
int infin0
eAt bici eAt dt
Non-Symmetric Cross Gramian
WZ =Msumi=1
Qsumj=1
int infin0
eAt bicj eAt dt
=
int infin0
eAt( Msumi=1
bi)( Qsum
j=1
cj)
eAt dt
Interestingly there exists a connection to tangential interpolation
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2142
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
emgr ndash Empirical Gramian Framework (Version 40)
Empirical Gramians
Empirical Controllability Gramian
Empirical Observability Gramian
Empirical Linear Cross Gramian
Empirical Cross Gramian
Empirical Sensitivity Gramian
Empirical Identifiability Gramian
Empirical Joint Gramian
Features
Interfaces for Solver Kernel Distributed Memory
Non-Symmetric option for all cross Gramians
Compatible with OCTAVE and MATLAB
Vectorized and parallelizable
Open-source licensed
More info at httpgramiande
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Combined Reduction(Together with M Ohlberger)
1 Empirical-Cross-Gramian-Based
2 Greedy-Optimization-Based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2342
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Iterative Projection Assembly [Lieberman et alrsquo10]
Idea Alternatingly extend parameter and state projection
1 Select parameter base vector θI+1 for parameter projectionΠI+1 = ΠI cup (θI+1 minus ΠIΠ
ᵀI θI+1)
2 Select state base vector x(θI+1) based on parameter θI+1UI+1 = UI cup (x(θI+1)minus UIU
TI x(θI+1))
3 Go to 1
Parameter selection Greedy method
State selection Energy-based
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2442
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Parameter Reduction [Bui-Thanh et alrsquo08]
Greedy Sampling Strategy
Locate currently worst approximated parameter
Formulation and solution as optimization problem
θI+1 = arg maxθisinΘ y(θ)minus y(ΠIΠᵀI θ)2
L2minus β2θ2
2
subject to
x(t) = f (t x(t) u(t) θ)
y(t) = g(t x(t) u(t) θ)
x(0) = x0
Adaptive sampling (no pre-defined grid)
Tikhonov regularization
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2542
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
State Reduction
In each iteration after selection of θI+1
1 Simulate a state trajectory x(θI+1)
2 Select principal component x(θI+1)
Based on input-to-state mapping u 7rarr x
U1 = arg minUisinRNtimesnUᵀU=1 x(θI+1)minus Uxr (θI+1)2L2
POD model reduction
proven method for nonlinear systems
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2642
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Data-Driven Regularization [Himpe amp Ohlbergerrsquo15]
An inverse problem provides data yd
Extended cost function with data mismatch as regularization
Jd = y(θ)minus yr (θr )2L2minus β2θ2
2 minus βdy(θ)minus yd2L2
Reduced order model is specific to data
Partial inversion during reduction
Accelerates projection assembly
Determining weighting coefficient more complicated
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2742
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Optimization-Based Combined Reduction
Truncated parameter-space projection
ΠI+1 rarr ΛI+1 = ΠᵀI+1
Truncated state-space projection
UI+1 rarr VI+1 = UᵀI+1
Greedy-optimization-based reduced order model
xr (t) = VI+1f (tUI+1xr (t) u(t)ΠI+1θr )
yr (t) = g(tUI+1xr (t) u(t)ΠI+1θr )
xr (0) = VI+1x0
θr = ΛI+1θ
Software Implementation httpgithubcomgramianoptmor
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2842
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Combined Reduction INonlinear Resistor-Capacitor Ladder Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 2942
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Combined Reduction IIHyperbolic Network Model Gramian- vs Optimization-Based
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
2040
6080
100
2040
6080
100
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3042
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Combined Reduction IIIfMRI amp fNIRS Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
1632
4864
80
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3142
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Combined Reduction IIIIEEG amp MEG Dynamic Causal Model Gramian- vs Optimization-Based
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
51102
153204
255
3264
96128
160
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Parame
ter
State
L2otimes L2-output error for varying reduced state and parameter dimensions
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Hierarchical Approximate POD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3342
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Hierarchical Approximate POD(Together with T Leibner amp S Rave)
Why HAPOD Memory-bound or compute-bound POD applications
Properties
Column-wise slicing of snapshotsMean L2 projection error boundMaximum mode boundRooted tree organizationModular POD backend
Special Cases
Distributed HAPOD (Minimum communication parallel)Rolling HAPOD (Minimum memory footprint)
See our preprint httparxivorgabs160705210
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3442
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
HAPODDefinitionLet S sub V be a finite multiset of snapshot vectors in a Hilbert space V Given arooted tree T and mappings
D S rarr LT εT NT rarr Rge0
define recursively for each α isin NT
HAPOD(S T D εT )(α) = POD(Iα εT (α))
where the local input data multiset Iα is given by
Iα =
Dminus1(α) α isin LT⋃
βisinCT (α)σn middot ϕn|(σn ϕn) isin HAPOD(S T D εT )(β) otherwise
We call HAPOD(S T D εT ) the hierarchical approximate POD (HAPOD) ofS for the rooted tree T the snapshot mapping D and the local tolerances εT
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3542
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Faster Empirical Cross Gramians
Simplified Description of the Empirical Cross Gramian
WX =Msum
m=1
int infin0
Ψm(t) dt isin RNtimesN
Ψmij (t) = 〈xmi (t) y jm(t)〉
Column-Wise Compution of the Empirical Cross Gramian (k-th column)
rArr WX lowastk =Msum
m=1
int infin0
ψmk(t) dt isin RNtimes1
ψmki (t) = 〈xmi (t) ykm(t)〉
Low-Rank Empircal Cross Gramian via HAPOD
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3642
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
MathEnergy Project(Together with P Benner and S Grundel)
MathEnergy ndash Mathematical Key Technologies for Evolving Energy Grids
Subproject Model Order Reduction
Model Reduction for Gas Networks
Software implementation
Power-to-Gas integration
BMWi funded
Federal Ministryfor Economic Affairsand Energy
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3742
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Model [Grundel et alrsquo14]
Hierarchical Model
1 Network
2 Pipes (Euler equation)
partp(x t)
partt= minusc2
A
partq(x t)
partxpartq(x t)
partt= minusApartp(x t)
partx+λc2
2DA
q(x t)|q(x t)|p(x t)
Challenges
Semi-discretized rarr DAE System
Nonlinear
Hyperbolic
Perspective Extensions Compressibility Elevation Temperature
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3842
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Reduction
Empirical Gramians
Modifications for
Hyperbolic systemsDAE systems
Alternative methods
Efficient simulation
FOMROM
Implementation
Comparison of
ModelsReduction approaches
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 3942
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
Scientific Sideshows
Replicability Reproducibility amp Reusability (doi103934Math20163261)
MORwiki (httpmodelreductionorg)
MOR Benchmark Framework
Stability Analysis on ROMs (with A Petersen)
RKHS Method for Empirical Gramians (with B Hamzi)
Scientific Computing on Single-Board Computers
FlexiBLAS (httpwwwmpi-magdeburgmpgdeprojectsflexiblas)
Explicit Runge-Kutta Methods
Efficient Octave MATLAB Code (httpgitiomtips)
ldquoGoodrdquo Colormaps
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4042
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
tldl
Combined State and Parameter Reduction
for Nonlinear Network Systems
using Empirical Cross Gramians
httphimpescience
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4142
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242
ReferencesC Himpe and T Leibner and S Rave Hierarchical Approximate Proper OrthogonalDecomposition Preprint arXiv mathNA 1ndash24 2016
J Fehr J Heiland C Himpe and J Saak Best Practices for ReplicabilityReproducibility and Reusability of Computer-Based Experiments Exemplified by ModelReduction Software AIMS Mathematics 1(3) 262ndash281 2016
C Himpe and M Ohlberger A Note on the Non-Symmetric Cross Gramian SystemScience and Control Engineering 4(1) 199ndash208 2016
U Baur P Benner B Haasdonk C Himpe I Martini and M Ohlberger Comparison ofmethods for parametric model order reduction of instationary problems To appear inldquoModel Reduction and Approximation Theory and Algorithmsrdquo 2016
C Himpe and M Ohlberger Data-driven combined state and parameter reduction forinverse problems Advances in Computational Mathematics (MoRePaS Special Issue)41(5)1343ndash1364 2015
C Himpe and M Ohlberger The Empirical Cross Gramian for Parametrized NonlinearSystems In MATHMOD 2015 vol 48(1) pages 727ndash728 2015
C Himpe and M Ohlberger Cross-Gramian Based Combined State and ParameterReduction for Large-Scale Control Systems Mathematical Problems in Engineering20141ndash13 2014
C Himpe and M Ohlberger Model Reduction for Complex Hyperbolic Networks InProceedings of the ECC pages 2739ndash2743 2014
C Himpe and M Ohlberger A Unified Software Framework for Empirical GramiansJournal of Mathematics 20131ndash6 2013
C Himpe himpempi-magdeburgmpgde Empirical Gramians and Friends 4242