MAX PLANCK INSTITUTE
FOR DYNAMICS OF COMPLEX
TECHNICAL SYSTEMS
MAGDEBURG
GAMM Workshop onApplied and Numerical Linear Algebra
with Special Emphasis on Model Reduction
September 22nd and 23rd , 2011
A Goal Oriented Low Rank Dual ADIIteration for Balanced Truncation
Jens Saakjoint work with
Peter Benner and Patrick Kurschner
Max Planck Institute for Dynamics of Complex Technical SystemsComputational Methods in Systems and Control Theory
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 1/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Outline
1 Balanced Truncation in a Nutshell
2 Solving Large Lyapunov Equations
3 Existing Approaches “Simultaneously” Treating the Dual LyapunovEquations
4 Dual LR-BT-ADI
5 Numerical Results
6 Conclusions and Future Perspectives
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 2/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Balanced Truncation in a NutshellBasic Idea
Idea:
The system Σ, in realization (A,B,C ), is called balanced, if thesolutions P,Q of the Lyapunov equations
AP + PAT + BBT = 0, ATQ + QA + CTC = 0,
satisfy: P = Q = diag(σ1, . . . , σn) where σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
{σ1, . . . , σn} are the Hankel singular values (HSVs) of Σ.
A balanced realization is computed via state space transformation
T : (A,B,C) 7→ (TAT−1,TB,CT−1)
=
([A11 A12
A21 A22
],
[B1
B2
],[C1 C2
]).
Truncation reduced order model: (A, B, C ) = (A11,B1,C1).
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 3/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Balanced Truncation in a NutshellBasic Idea
Idea:
The system Σ, in realization (A,B,C ), is called balanced, if thesolutions P,Q of the Lyapunov equations
AP + PAT + BBT = 0, ATQ + QA + CTC = 0,
satisfy: P = Q = diag(σ1, . . . , σn) where σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
{σ1, . . . , σn} are the Hankel singular values (HSVs) of Σ.
A balanced realization is computed via state space transformation
T : (A,B,C) 7→ (TAT−1,TB,CT−1)
=
([A11 A12
A21 A22
],
[B1
B2
],[C1 C2
]).
Truncation reduced order model: (A, B, C ) = (A11,B1,C1).
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 3/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Balanced Truncation in a NutshellBasic Idea
Idea:
The system Σ, in realization (A,B,C ), is called balanced, if thesolutions P,Q of the Lyapunov equations
AP + PAT + BBT = 0, ATQ + QA + CTC = 0,
satisfy: P = Q = diag(σ1, . . . , σn) where σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
{σ1, . . . , σn} are the Hankel singular values (HSVs) of Σ.
A balanced realization is computed via state space transformation
T : (A,B,C) 7→ (TAT−1,TB,CT−1)
=
([A11 A12
A21 A22
],
[B1
B2
],[C1 C2
]).
Truncation reduced order model: (A, B, C ) = (A11,B1,C1).
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 3/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Balanced Truncation in a NutshellBasic Idea
Idea:
The system Σ, in realization (A,B,C ), is called balanced, if thesolutions P,Q of the Lyapunov equations
AP + PAT + BBT = 0, ATQ + QA + CTC = 0,
satisfy: P = Q = diag(σ1, . . . , σn) where σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
{σ1, . . . , σn} are the Hankel singular values (HSVs) of Σ.
A balanced realization is computed via state space transformation
T : (A,B,C) 7→ (TAT−1,TB,CT−1)
=
([A11 A12
A21 A22
],
[B1
B2
],[C1 C2
]).
Truncation reduced order model: (A, B, C ) = (A11,B1,C1).
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 3/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Balanced Truncation in a NutshellImplementation
The SR Method1 Compute (Cholesky)factors of the solutions to the Lyapunov
equation,P = STS , Q = RTR.
2 Compute singular value decomposition
SRT = [U1, U2 ]
[Σ1
Σ2
] [V T1
V T2
].
3 DefineW := RTV1Σ
−1/21 , V := STU1Σ
−1/21 .
4 Then the reduced order model is (W TAV ,W TB,CV ).
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 4/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Balanced Truncation in a NutshellImplementation
The SR Method1 Compute (Cholesky)factors of the solutions to the Lyapunov
equation,P = STS , Q = RTR.
2 Compute singular value decomposition
SRT = [U1, U2 ]
[Σ1
Σ2
] [V T1
V T2
].
3 DefineW := RTV1Σ
−1/21 , V := STU1Σ
−1/21 .
4 Then the reduced order model is (W TAV ,W TB,CV ).
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 4/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Balanced Truncation in a NutshellImplementation
The SR Method1 Compute (Cholesky)factors of the solutions to the Lyapunov
equation,P = STS , Q = RTR.
2 Compute singular value decomposition
SRT = [U1, U2 ]
[Σ1
Σ2
] [V T1
V T2
].
3 DefineW := RTV1Σ
−1/21 , V := STU1Σ
−1/21 .
4 Then the reduced order model is (W TAV ,W TB,CV ).
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 4/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Solving Large Lyapunov EquationsLRCF-ADI
Lyapunov equation
Consider
FX + XFT = −GGT
with F ∈ Rn×n Hurwitz, G ∈ Rn×m, m� n.
Observation in practice:[Penzl ’99, Ant./Sor./Zhou ’02, Grasedyck ’04]
rank(X , τ) = r � n
⇒ Compute low-rank solution factorZ ∈ Rn×r , r � n.
X ≈ Z ZT . 50 100 2000 250010−25
10−13
10−1
· · ·
u
singular values for n = 2 500 example
σj (X )
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 5/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Solving Large Lyapunov EquationsLRCF-ADI e.g.,[Benner/Li/Penzl ’08]
Consider FX + XFT = −GGT F ∈ Rn×n,G ∈ Rn×p
Task Find Z ∈ Cn,nz , such that nz � n and X ≈ ZZH
Algorithm
V1 =√−2Re p1(F + p1I )
−1G , Z1 = V1
Vi =
√Re pi√
Re pi−1
[I − (pi + pi−1)(F + pi I )
−1]Vi−1 Zi = [Zi−1Vi ]
For certain shift parameters {p1, ..., pJ} ⊂ C<0.
Stop if
‖ViVHi ‖ is small, or
‖FZiZHi + ZiZ
Hi FT + GGT‖ is small.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 6/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Solving Large Lyapunov EquationsLRCF-ADI e.g.,[Benner/Li/Penzl ’08]
Consider FX + XFT = −GGT F ∈ Rn×n,G ∈ Rn×p
Task Find Z ∈ Cn,nz , such that nz � n and X ≈ ZZH
Algorithm
V1 =√−2Re p1(F + p1I )
−1G , Z1 = V1
Vi =
√Re pi√
Re pi−1
[I − (pi + pi−1)(F + pi I )
−1]Vi−1 Zi = [Zi−1Vi ]
For certain shift parameters {p1, ..., pJ} ⊂ C<0.
Stop if
‖ViVHi ‖ is small, or
‖FZiZHi + ZiZ
Hi FT + GGT‖ is small.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 6/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Solving Large Lyapunov EquationsG-LRCF-ADI (E invertible) e.g.,[S. ’09]
Consider FXET + EXFT = −GGT E ,F ∈ Rn×n,G ∈ Rn×p
Task Find Z ∈ Cn,nz , such that nz � n and X ≈ ZZH
Algorithm
V1 =√−2Re p1(F + p1E)−1G , Z1 = V1
Vi =
√Re pi√
Re pi−1
[I − (pi + pi−1)(F + piE)−1
]EVi−1 Zi = [Zi−1Vi ]
For certain shift parameters {p1, ..., pJ} ⊂ C<0.
Stop if
‖ViVHi ‖ is small, or
‖FZiZHi ET + EZiZ
Hi FT + GGT‖ is small.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 6/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Solving Large Lyapunov EquationsS-LRCF-ADI ((E ,F ) index 1) e.g.,[Rommes/Freitas/Martins ’08]
Consider FXET11 + E 11XFT = −G GT E 11, F ∈ Rn×n, G ∈ Rn×p
Task Find Z ∈ Cn,nz , such that nz � n and X ≈ ZZH
Algorithm[V1
∗
]=√−2Re p1
[F11 + p1E11 F12
F21 F22
]−1 [B1
B2
], Z1 = V1[
Vi
∗
]=
√Re pi√
Re pi−1
[I − (pi + pi−1)
[F11 + piE11 F12
F21 F22
]−1] [E11Vi−1
0
]Zi = [Zi−1Vi ]
For certain shift parameters {p1, ..., pJ} ⊂ C<0.
Stop if
‖ViVHi ‖ is small, or
‖FZiZHi ET
11 + E 11ZiZHi FT + G GT‖ is small.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 6/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Solving Large Lyapunov EquationsS-LRCF-ADI e.g.,[Rommes/Freitas/Martins ’08]
Consider FXET11 + E 11XFT = −G GT E 11, F ∈ Rn×n, G ∈ Rn×p
Task Find Z ∈ Cn,nz , such that nz � n and X ≈ ZZH
Algorithm[V1
∗
]=√−2Re p1
[F11 + p1E11 F12
F21 F22
]−1 [B1
B2
], Z1 = V1[
Vi
∗
]=
√Re pi√
Re pi−1
[I − (pi + pi−1)
[F11 + piE11 F12
F21 F22
]−1] [E11Vi−1
0
]Zi = [Zi−1Vi ]
Can ensure Z ∈ Rn,nz even if {p1, ..., pJ} 6⊂ Rsee next talk by Patrick Kurschner
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 6/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Solving Large Lyapunov EquationsDrawbacks of LR-ADI Based BT
Performing LR-ADI based BT is computationally expensivecompared to simple Krylov based approaches.
Main effort is solving the dual Lyapunov equations.
Central Question
Can we reduce the time for computingthe balancing transformations?
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 7/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Existing Approaches “Simultaneously” Treating theDual Lyapunov EquationsCross Gramian based Methods e.g. [Fernando/Nicholson ’83; Sorensen/Antoulas ’02]
IdeaSolve AX + XA = −BC for the cross gramian X ,
Proceed with knowledge X 2 = PQ, i.e., for the HSV σi = |λj(X )|.
e.g., repeatedly solve projected equations of the form
+ = for
Requires symmetric system or symmetrizer Ψ, such that
AΨ = ΨA∗, B = ΨC∗,
Ψ may be expensive to compute
number of inputs and outputs must coincide.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 8/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Existing Approaches “Simultaneously” Treating theDual Lyapunov EquationsApproximate Implicit Subspace Iteration with Alternating Directions (AISIAD)[Zhou/Sorensen ’08]
Idea
Solve AP +PAT = −BBT and ATQ +QA = −CTC simultaneously.
Alternate the projection spaces.
alternatingly solve 2 equations of the form
+ = + =
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 9/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Existing Approaches “Simultaneously” Treating theDual Lyapunov EquationsApproximate Implicit Subspace Iteration with Alternating Directions (AISIAD)[Zhou/Sorensen ’08]
Idea
Solve AP +PAT = −BBT and ATQ +QA = −CTC simultaneously.
Alternate the projection spaces.
alternatingly solve 2 equations of the form
+ = + =
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 9/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Existing Approaches “Simultaneously” Treating theDual Lyapunov EquationsApproximate Implicit Subspace Iteration with Alternating Directions (AISIAD)[Zhou/Sorensen ’08]
Idea
Solve AP +PAT = −BBT and ATQ +QA = −CTC simultaneously.
Alternate the projection spaces.
alternatingly solve 2 equations of the form
+ = + =
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 9/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Existing Approaches “Simultaneously” Treating theDual Lyapunov EquationsApproximate Implicit Subspace Iteration with Alternating Directions (AISIAD)[Zhou/Sorensen ’08]
Idea
Solve AP +PAT = −BBT and ATQ +QA = −CTC simultaneously.
Alternate the projection spaces.
alternatingly solve 2 equations of the form
+ = + =
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 9/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Existing Approaches “Simultaneously” Treating theDual Lyapunov EquationsApproximate Implicit Subspace Iteration with Alternating Directions (AISIAD)[Zhou/Sorensen ’08]
Idea
Solve AP +PAT = −BBT and ATQ +QA = −CTC simultaneously.
Alternate the projection spaces.
alternatingly solve 2 equations of the form
+ = + =
After convergence use and to compute balancing transformations.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 9/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Existing Approaches “Simultaneously” Treating theDual Lyapunov EquationsApproximate Implicit Subspace Iteration with Alternating Directions (AISIAD)[Zhou/Sorensen ’08]
Idea
Solve AP +PAT = −BBT and ATQ +QA = −CTC simultaneously.
Alternate the projection spaces.
Initial projection basis required.
2 growingly expensive linear equations per step.
Reuse of solvers for the step equations?
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 9/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Dual LR-BT-ADISimultaneous Solution of the Dual Lyapunov Equations
Key Target
Reuse LU-factorizations in a sparse direct solver framework.
Vi =
√Re pi√Re pi−1
[I − (pi + pi−1)(F + piE )−1
]EVi−1,
LU := (F + piE ) ⇒ UHLH = (FT + piET )
complex shifts come in conjugate pair,
they are used one after another,
reverse order of complex conjugate shifts for the second equation.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 10/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Dual LR-BT-ADISimultaneous Solution of the Dual Lyapunov Equations
Key Target
Reuse LU-factorizations in a sparse direct solver framework.
Vi =
√Re pi√Re pi−1
[I − (pi + pi−1)(F + piE )−1
]EVi−1,
LU := (F + piE ) ⇒ UHLH = (FT + piET )
complex shifts come in conjugate pair,
they are used one after another,
reverse order of complex conjugate shifts for the second equation.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 10/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Dual LR-BT-ADISimultaneous Solution of the Dual Lyapunov Equations
Key Target
Reuse LU-factorizations in a sparse direct solver framework.
Vi =
√Re pi√Re pi−1
[I − (pi + pi−1)U−1L−1
]EVi−1,
Wi =
√Re pi√Re pi−1
[I − (pi + pi−1)L−HU−H
]ETWi−1.
complex shifts come in conjugate pair,
they are used one after another,
reverse order of complex conjugate shifts for the second equation.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 10/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Dual LR-BT-ADISimultaneous Solution of the Dual Lyapunov Equations
Key Target
Reuse LU-factorizations in a sparse direct solver framework.
Vi =
√Re pi√Re pi−1
[I − (pi + pi−1)U−1L−1
]EVi−1,
Wi =
√Re pi√Re pi−1
[I − (pi + pi−1)L−HU−H
]ETWi−1.
complex shifts come in conjugate pair,
they are used one after another,
reverse order of complex conjugate shifts for the second equation.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 10/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Dual LR-BT-ADIStopping the Iteration
ProblemNumber of columns in LRCFs limits ROM dimension.
Lyapunov residuals usually totally unrelated to ROM quality.
Idea
Use goal oriented stopping criteria.
Identify and monitor the property of interest to underlying application.
In Balanced Truncation MOR:
Assume we are interested in an order k ROM.
Monitor the relative change of k leading HSVs.
Stop when leading HSVs do no longer change.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 11/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Dual LR-BT-ADIStopping the Iteration
ProblemNumber of columns in LRCFs limits ROM dimension.
Lyapunov residuals usually totally unrelated to ROM quality.
Idea
Use goal oriented stopping criteria.Identify and monitor the property of interest to underlying application.
In Balanced Truncation MOR:
Assume we are interested in an order k ROM.
Monitor the relative change of k leading HSVs.
Stop when leading HSVs do no longer change.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 11/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Dual LR-BT-ADIStopping the Iteration
ProblemNumber of columns in LRCFs limits ROM dimension.
Lyapunov residuals usually totally unrelated to ROM quality.
Idea
Use goal oriented stopping criteria.Identify and monitor the property of interest to underlying application.
In Balanced Truncation MOR:
Assume we are interested in an order k ROM.
Monitor the relative change of k leading HSVs.
Stop when leading HSVs do no longer change.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 11/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Dual LR-BT-ADIStopping the Iteration
ProblemNumber of columns in LRCFs limits ROM dimension.
Lyapunov residuals usually totally unrelated to ROM quality.
Idea
Use goal oriented stopping criteria.Identify and monitor the property of interest to underlying application.
In Balanced Truncation MOR:
Assume we are interested in an order k ROM.
Monitor the relative change of k leading HSVs.
Stop when leading HSVs do no longer change.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 11/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsTest Examples and Hardware
Rail Model [Benner/S. ’03-’05]
Oberwolfach Collection Rail Coolinga
n = 1357
inputs 7, outputs 6
ahttp://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/
Steel\Profiles\(38881)
BIPS07 [Freitas/Rommes/Martins ’08]
index 1 descriptor model of a power networkb
n = 13 275, nE11 = 1693
inputs 4, outputs 4
condest(A22) = 8.275 · 1026
bhttp://sites.google.com/site/rommes/software/bips07_1693.mat
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 12/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsTest Examples and Hardware
Hardware
dual Intel®Xeon®W3503 @ 2.40GHz
Cache Size: 4MB
RAM: 6GB
SofwareOS: Ubuntu Linux 10.04LTS
32bit kernel 2.6.32-25-generic-pae
MATLAB® 7.11.0 (R2010b)
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 13/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 28
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 29
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 30
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 31
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 32
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 33
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 34
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 35
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 36
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 37
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 38
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 39
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 40
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 41
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 42
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 43
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 44
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 45
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 46
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 47
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 48
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 49
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 50
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 51
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 52
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 53
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 54
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 55
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 56
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 57
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 58
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 59
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−22
10−17
10−12
10−7
10−2
i
σi
Leading Hankel Singular Values ( σiσmax≥ eps)
Original
dualADI
Iteration: 60
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 14/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 29
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 30
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 31
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 32
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 33
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 34
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 35
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 36
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 37
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 38
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 39
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 40
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 41
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 42
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 43
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 44
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 45
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 46
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 47
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 48
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 49
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 50
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 51
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 52
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 53
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 54
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 55
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 56
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 57
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 58
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 59
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
20 40 60 80 100 120 140 16010−24
10−18
10−12
10−6
100
i
|σ−σ(old) |
σmax
relative change of HSVs
Iteration: 60
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 15/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsRail Model
ObservationLeading HSVs converge from below.
Qualitative explanation:
LRCF-ADI is a shifted rational Arnoldi process.
HSVs are thus approximated via Ritz values.
Ritz values converge to the eigenvalues from the inside of thespectrum.
Quantitative analysis: work in progress.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 16/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
Test Setup
We take the same settings as in [Freitas/Rommes/Martins ’08], i.e.:
shift parameters
50 Penzl shiftsfrom 80 Ritz values and 80 harmonic Ritz values
reduced order model dimension 32
maximum iteration number 80
α-shifted S-LRCF-ADI with α = 0.05
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 17/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
5 10 15 20 25 3010−1
100
101
102
i
σi
Leading 32 Hankel Singular Values
Original
S-LRCF-ADI
dual-S-LRCF-ADI
Do not trust residuals!
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 18/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
10−2 10−1 100 101 102100
101
102
103
ω
σmax(G
(ω
))
Transfer functions of original and reduced systems
original system
reduced system
dual reduced
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 19/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
10−2 10−1 100 101 10210−1
100
101
102
ω
σmax(G
(ω
)−
Gr(ω
))
absolute model reduction error
2 S-LRCF-ADIs
dual-S-LRCF-ADI
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 20/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
10−2 10−1 100 101 10210−3
10−2
10−1
100
ω
σmax(G
(ω)−
Gr(ω))
σmax(G
(ω))
relative model reduction error
2 S-LRCF-ADIs
dual-S-LRCF-ADI
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 21/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
10 20 30 40 50 60 70 8010−8
10−7
10−6
10−5
10−4
niter
res(niter
)re
s 0Residual History for S-LRCF-ADI
Controllability
Observability
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 22/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
5 10 15 20 25 30
10−1
100
101
102
i
σi
Leading 32 Hankel Singular Values
Original
S-LRCF-ADI(80)
dual-S-LRCF-ADI(80)
S-LRCF-ADI(15)
Do not trust residuals!
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 23/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
5 10 15 20 25 30
10−1
100
101
102
i
σi
Leading 32 Hankel Singular Values
Original
S-LRCF-ADI(80)
dual-S-LRCF-ADI(80)
S-LRCF-ADI(15)
dual-S-LRCF-ADI(250)
Do not trust residuals!
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 23/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
maxiter 50 75 100 125 150
tdual in s 2.97 5.22 7.31 10.33 14.72tS-LRCF-ADI in s 19.87 27.82 37.03 47.65 58.97
speedup 6.68 5.33 5.06 4.61 4.01
change HSV 2.151e-1 1.123e-1 7.440e-2 6.579e-2 6.438e-2final res. B 3.237e-5 3.237e-5 3.237e-5 3.237e-5 3.237e-5final res. C 5.513e-8 6.194e-8 5.900e-8 5.778e-8 5.710e-8
175 200 225 250
20.20 29.99 39.34 52.5172.16 86.32 98.50 111.7
3.57 2.88 2.50 2.13
6.389e-2 6.282e-2 6.189e-2 6.058e-23.237e-5 3.237e-5 3.237e-5 3.237e-55.703e-8 5.706e-8 5.707e-8 5.710e-8
Column compression via RRQR can reduce the SVD cost in the dual method:
maxiter = 250, CC-frequency=20, RRQR-tolerance=eps tdual ≈ 37s
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 24/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Numerical ResultsBIPS07
maxiter 50 75 100 125 150
tdual in s 2.97 5.22 7.31 10.33 14.72tS-LRCF-ADI in s 19.87 27.82 37.03 47.65 58.97
speedup 6.68 5.33 5.06 4.61 4.01
change HSV 2.151e-1 1.123e-1 7.440e-2 6.579e-2 6.438e-2final res. B 3.237e-5 3.237e-5 3.237e-5 3.237e-5 3.237e-5final res. C 5.513e-8 6.194e-8 5.900e-8 5.778e-8 5.710e-8
175 200 225 250
20.20 29.99 39.34 52.5172.16 86.32 98.50 111.7
3.57 2.88 2.50 2.13
6.389e-2 6.282e-2 6.189e-2 6.058e-23.237e-5 3.237e-5 3.237e-5 3.237e-55.703e-8 5.706e-8 5.707e-8 5.710e-8
Column compression via RRQR can reduce the SVD cost in the dual method:
maxiter = 250, CC-frequency=20, RRQR-tolerance=eps tdual ≈ 37s
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 24/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Conclusions and Future Perspectives
ConclusionsSimultaneous handling of the dual Lyapunov equations candrastically improve the performance.
Goal oriented stopping criteria enhance the reliability of the results.
Future WorkQuantify convergence results.
Investigate reliable adaptive ROM dimension control.
Adapt stopping criterion to finite arithmetic.
Thank you very muchfor listening.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 25/25
BT in a Nut ∗LRCF-ADI Existing Approaches Dual LR-BT-ADI Numerical Results Conclusions
Conclusions and Future Perspectives
ConclusionsSimultaneous handling of the dual Lyapunov equations candrastically improve the performance.
Goal oriented stopping criteria enhance the reliability of the results.
Future WorkQuantify convergence results.
Investigate reliable adaptive ROM dimension control.
Adapt stopping criterion to finite arithmetic.
Thank you very muchfor listening.
Max Planck Institute Magdeburg Jens Saak, Dual ADI for Balanced Truncation 25/25