MAX PLANCK INSTITUTE
FOR DYNAMICS OF COMPLEX
TECHNICAL SYSTEMS
MAGDEBURG
Workshop on Model Order Reductionof Transport-dominated Phenomena
Berlin, May 19–20, 2015
Model Order Reduction of ParametrizedNonlinear Evolution Equations with
Applications in Chromatography
Peter Benner
Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburg, Germany
Email: [email protected]
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 1/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Outline
1 MotivationGeneral set-up: nonlinear parametric systemsMotivating Examples
2 Model Order ReductionPetrov-Galerkin ProjectionEmpirical Interpolation Method
3 Error BoundPrimal-only Error BoundPrimal-dual Output Error Bound
4 Basis Construction and Adaptive Snapshot SelectionPOD-Greedy AlgorithmAdaptive Snapshot Selection
5 Numerical Results
6 Conclusions and Outlook
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 2/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Collaborators
Lihong Feng Yongjin Zhang AndreasSeidel-Morgenstern
MPI MagdeburgComputational Methods in Systems and Control (CSC)
Physical and Chemical Foundations of Process Engineering (PCF)
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 3/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationGeneral set-up: nonlinear parametric systems
Nonlinear Parametric Systems
E (t, µ)dx
dt= A(t, µ)x + f (x , µ),
or
E (tk , µ)xk+1 = A(tk , µ)xk + f (xk , µ),
x , xk ∈ Wn ⊂ Rn, E ,A ∈ Rn×n, n is large.
Often, the output y = g(x), or y = Cx , is of interest quantities-of-interest.
Multi-query context:Solve the ODE system for many varying values of µ ∈ Ω ⊂ Rd , e.g.,optimization, real-time control, inverse problems, . . .
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 4/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationMotivating Example: Batch Chromatography
Column
ba
Solvent
Pump
a + b
b
a
b a
t
int
cyct
Q 1t 2t 3t 4t
Feed
t
Products
concentration
concentration
Principle of batch chromatography for binary separation.
∂cz
∂t+
1− εε
∂qz
∂t= −∂cz
∂x+
1
Pe
∂2cz
∂x2, 0 < x < 1,
∂qz
∂t=
L
Q/(εAc )κz (qEq
z − qz ), 0 ≤ x ≤ 1,z = a, b
A convection-dominated system, the Peclet number Pe is large.
Requires long-time integration process.
A nonlinear parametric coupled system, parameters µ := (Q, tin).
What are the optimal operating conditions? PDE constrained optimization.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 5/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationMotivating Example: Batch Chromatography
Column
ba
Solvent
Pump
a + b
b
a
b a
t
int
cyct
Q 1t 2t 3t 4t
Feed
t
Products
concentration
concentration
Principle of batch chromatography for binary separation.∂cz
∂t+
1− εε
∂qz
∂t= −∂cz
∂x+
1
Pe
∂2cz
∂x2, 0 < x < 1,
∂qz
∂t=
L
Q/(εAc )κz (qEq
z − qz ), 0 ≤ x ≤ 1,z = a, b
A convection-dominated system, the Peclet number Pe is large.
Requires long-time integration process.
A nonlinear parametric coupled system, parameters µ := (Q, tin).
What are the optimal operating conditions? PDE constrained optimization.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 5/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationMotivating Example: Batch Chromatography
Column
ba
Solvent
Pump
a + b
b
a
b a
t
int
cyct
Q 1t 2t 3t 4t
Feed
t
Products
concentration
concentration
Principle of batch chromatography for binary separation.∂cz
∂t+
1− εε
∂qz
∂t= −∂cz
∂x+
1
Pe
∂2cz
∂x2, 0 < x < 1,
∂qz
∂t=
L
Q/(εAc )κz (qEq
z − qz ), 0 ≤ x ≤ 1,z = a, b
A convection-dominated system, the Peclet number Pe is large.
Requires long-time integration process.
A nonlinear parametric coupled system, parameters µ := (Q, tin).
What are the optimal operating conditions? PDE constrained optimization.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 5/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationMotivating Example: Batch Chromatography
Column
ba
Solvent
Pump
a + b
b
a
b a
t
int
cyct
Q 1t 2t 3t 4t
Feed
t
Products
concentration
concentration
Principle of batch chromatography for binary separation.∂cz
∂t+
1− εε
∂qz
∂t= −∂cz
∂x+
1
Pe
∂2cz
∂x2, 0 < x < 1,
∂qz
∂t=
L
Q/(εAc )κz (qEq
z − qz ), 0 ≤ x ≤ 1,z = a, b
A convection-dominated system, the Peclet number Pe is large.
Requires long-time integration process.
A nonlinear parametric coupled system, parameters µ := (Q, tin).
What are the optimal operating conditions? PDE constrained optimization.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 5/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationMotivating Example: Batch Chromatography
Column
ba
Solvent
Pump
a + b
b
a
b a
t
int
cyct
Q 1t 2t 3t 4t
Feed
t
Products
concentration
concentration
Principle of batch chromatography for binary separation.∂cz
∂t+
1− εε
∂qz
∂t= −∂cz
∂x+
1
Pe
∂2cz
∂x2, 0 < x < 1,
∂qz
∂t=
L
Q/(εAc )κz (qEq
z − qz ), 0 ≤ x ≤ 1,z = a, b
A convection-dominated system, the Peclet number Pe is large.
Requires long-time integration process.
A nonlinear parametric coupled system, parameters µ := (Q, tin).
What are the optimal operating conditions? PDE constrained optimization.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 5/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationMotivating Example: Simulated Moving Bed (SMB) Chromatography
SMB chromatographic process with 4 zones and 8 columns.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 6/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationMotivating Example: SMB Chromatography
SMB chromatographic process with 4 zones and 8 columns.
Governing equations are similar, but:
• More parameters, µ := (m1, . . . ,m4,QF)
• Multi-switching system
• Cyclic steady state computation
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 7/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationMotivating Example: SMB Chromatography
SMB chromatographic process with 4 zones and 8 columns.
Governing equations are similar, but:
• More parameters, µ := (m1, . . . ,m4,QF)
• Multi-switching system
• Cyclic steady state computation
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 7/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Motivation4-column SMB plant at MPI Magdeburg
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 8/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationSMB Chromatography — a practical application
Purified ArtemisininArtemisinin is the basic compound forproducing the malaria medicationArtesunate.
New SMB-based process developed atMPI Magdeburg (PCF roup) yields 99.5%purity (exceeding the limits set by WHOand FDA), based on new synthesis processinvented by Peter Seeberger (MPI Colloidsand Interfaces, Potsdam).
Process can be easily implemented inlow-cost plants in the countries where theplant Artemisia annua grows, mostly, inEast Asia.
Model plant built in Vietnam.
Much cheaper than current anti-Malariamedication, and much higher degree ofpurity!
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 9/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MotivationSMB Chromatography — a practical application
Purified Artemisinin
Seeberger andSeidel-Morgenstern wereawarded theHumanity in Science Prize 2015for this.
Artemisinin is the basic compound forproducing the malaria medicationArtesunate.
New SMB-based process developed atMPI Magdeburg (PCF roup) yields 99.5%purity (exceeding the limits set by WHOand FDA), based on new synthesis processinvented by Peter Seeberger (MPI Colloidsand Interfaces, Potsdam).
Process can be easily implemented inlow-cost plants in the countries where theplant Artemisia annua grows, mostly, inEast Asia.
Model plant built in Vietnam.
Much cheaper than current anti-Malariamedication, and much higher degree ofpurity!
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 9/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MOR for Nonlinear Parametric Systems
Original full order system (FOM)
E (t, µ)dx
dt= A(t, µ)x + f (x , µ),
orE (tk , µ)xk+1 = A(tk , µ)xk + f (xk , µ),
x , xk ∈ Wn ⊂ Rn, E ,A ∈ Rn×n, n is large.
Often the output y = g(x), or y = Cx is of interest.
Reduced-order model (ROM)
E (t, µ)dz
dt= A(t, µ)z + W T f (Vz , µ), x := Vz ,
orE (tk , µ)zk+1 = A(tk , µ)zk + W T f (Vzk , µ) xk := Vzk ,
E = W TEV , A = W TAV , W ,V ∈ Rn×N , z , zk ∈ RN , N n.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 10/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MOR for Nonlinear Parametric Systems
Original full order system (FOM)
E (t, µ)dx
dt= A(t, µ)x + f (x , µ),
orE (tk , µ)xk+1 = A(tk , µ)xk + f (xk , µ),
x , xk ∈ Wn ⊂ Rn, E ,A ∈ Rn×n, n is large.
Often the output y = g(x), or y = Cx is of interest.
Reduced-order model (ROM)
E (t, µ)dz
dt= A(t, µ)z + W T f (Vz , µ), x := Vz ,
orE (tk , µ)zk+1 = A(tk , µ)zk + W T f (Vzk , µ) xk := Vzk ,
E = W TEV , A = W TAV , W ,V ∈ Rn×N , z , zk ∈ RN , N n.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 10/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Remarks
Let y(t, µ) be the approximate output of interest. Arising questions are:
1 How to deal with the nonlinearity and/or non-affinity, i.e., efficientlycompute W T f (Vz , µ) or W T f (Vzk , µ)? EIM.
2 How to estimate the error in the quantities-of-interest, i.e.,‖y − y‖ ≤ ? Output error bound.
3 How to efficiently construct the projection matrices V and W ? Adaptive snapshot selection.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 11/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Remarks
Let y(t, µ) be the approximate output of interest. Arising questions are:
1 How to deal with the nonlinearity and/or non-affinity, i.e., efficientlycompute W T f (Vz , µ) or W T f (Vzk , µ)? EIM.
2 How to estimate the error in the quantities-of-interest, i.e.,‖y − y‖ ≤ ? Output error bound.
3 How to efficiently construct the projection matrices V and W ? Adaptive snapshot selection.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 11/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Remarks
Let y(t, µ) be the approximate output of interest. Arising questions are:
1 How to deal with the nonlinearity and/or non-affinity, i.e., efficientlycompute W T f (Vz , µ) or W T f (Vzk , µ)? EIM.
2 How to estimate the error in the quantities-of-interest, i.e.,‖y − y‖ ≤ ? Output error bound.
3 How to efficiently construct the projection matrices V and W ? Adaptive snapshot selection.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 11/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Empirical Interpolation Method (EIM)
Idea: construct a basis of interpolation functions (vectors), and use anaffine expression to approximate W T f (Vz , µ), i.e.,
W T f (Vz , µ) ≈ W TU︸ ︷︷ ︸Precomputed
β(z , µ).
Different methods have been proposed to construct the basis U ∈ Rn×M
and the corresponding coefficients β(z , µ):
Empirical interpolation method (EIM)[Barrault/Maday/Nguyen/Patera ’04]
Missing point estimation (MPE)[Astrid/Weiland/Willcox/Backx ’08, Faßbender/Vendl ’11]
Discrete empirical interpolation method (DEIM)[Chaturantabut/Sorensen ’10]
Empirical operator interpolation [Drohmann/Haasdonk/Ohlberger ’12]
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 12/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MOR for Nonlinear Parametric Systems
Original full order system (FOM)
E (t, µ)dx
dt= A(t, µ)x + f (x , µ),
orE (tk , µ)xk+1 = A(tk , µ)xk + f (xk , µ).
Reduced-order model (ROM)
E (t, µ)dz
dt= A(t, µ)z + W T f (Vz , µ), x := Vz ,
orE (tk , µ)zk+1 = A(tk , µ)zk + W T f (Vzk , µ) xk := Vzk ,
E = W TEV , A = W TAV , W ,V ∈ Rn×N , z , zk ∈ RN , N n.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 13/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
MOR for Nonlinear Parametric Systems
Original full order system (FOM)
E (t, µ)dx
dt= A(t, µ)x + f (x , µ),
orE (tk , µ)xk+1 = A(tk , µ)xk + f (xk , µ).
Use Empirical Interpolation to efficiently compute W T f (Vz , µ)
E (t, µ)dz
dt= A(t, µ)z + W TUβ(z , µ),
orE (tk , µ)zk+1 = A(tk , µ)zk + W TUβk (z , µ).
The fast computation can be achieved by the strategy of offline-onlinedecomposition, i.e., E , A and W TU can be precomputed once V ,W ,Uare obtained.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 13/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Error Bound
Consider the evolution scheme,
E (tk , µ)xk+1 = A(tk , µ)xk + f (xk , µ),yk+1 = Cxk+1.
The reduced-order model (ROM):
E (tk , µ)zk+1 = A(tk , µ)zk + W T f (Vzk , µ),yk+1 = CVzk+1.
Here, E (tk , µ) = W TE (tk , µ)V , A(tk , µ) = W TA(tk , µ)V , xk := Vzk
approximates xk , k = 0, . . . ,Tn.
Define the residual:
rk+1(µ) := A(tk , µ)xk + f (xk , µ)− E (tk , µ)xk+1.
We have the following error estimations.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 14/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-only Error BoundField Variable Error Bound
Theorem (Error Bound 1)[Drohmann/Haasdonk/Ohlberger ’12, Zhang/Feng/Li/Benner ’14]
Let ek (µ) := xk − xk and ekO(µ) := yk − yk be the error for the solution
and the output at time step tk , respectively. Under certain assumptions,we have:
‖e1(µ)‖ ≤ η1N,M (µ) := R
(0)F,µ,
‖ek (µ)‖ ≤ ηkN,M (µ) := R
(k−1)F,µ +
k−2∑i=0
k−1∏j=i+1
G(j)F,µ
R(i)F,µ, k = 2, . . . ,Tn.
where
R(i)F,µ =
∥∥E (t i , µ)−1r i+1(µ)∥∥ , i = 0, . . . , k − 1,
G(j)F,µ =
∥∥E (t j , µ)−1A(t j , µ)∥∥+ Lf
∥∥E (t j , µ)−1∥∥ , j = i + 1, . . . , k − 1.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 15/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-only Error Bound (Cont.)Output Error Bound
Theorem (Output Error Bound 1) [Zhang/Feng/Li/Benner ’14]
Under the assumptions of Prop. 1, we have:
‖ek+1O (µ)‖ ≤ G
(k)O,µη
kN,M (µ) + ||C ||||E (tk , µ)−1rk+1(µ)||,
whereG
(k)O,µ = ||CE (tk , µ)−1A(tk , µ)||+ Lf ||CE (tk , µ)−1||.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 16/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound
”Dual” system and the reduced ”dual” system
E (tk , µ)T xk+1du = −CT , W T
duE (tk , µ)TVduzk+1du = −W T
duCT .
Here, xkdu := Vduz
kdu approximates xk
du, k = 1, . . . ,Tn.
Residual of the reduced dual system:
rk+1du (µ) := −CT − E (tk , µ)T xk+1
du .
Recall residual of the ROM:
rk+1(µ) := A(tk , µ)xk + f (xk , µ)− E (tk , µ)xk+1.
Define an auxiliary vector,
rk+1(µ) := A(tk , µ)xk + f (xk , µ)− E (tk , µ)xk+1
= E (tk , µ)xk+1 − E (tk , µ)xk+1.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 17/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound
”Dual” system and the reduced ”dual” system
E (tk , µ)T xk+1du = −CT , W T
duE (tk , µ)TVduzk+1du = −W T
duCT .
Here, xkdu := Vduz
kdu approximates xk
du, k = 1, . . . ,Tn.
Residual of the reduced dual system:
rk+1du (µ) := −CT − E (tk , µ)T xk+1
du .
Recall residual of the ROM:
rk+1(µ) := A(tk , µ)xk + f (xk , µ)− E (tk , µ)xk+1.
Define an auxiliary vector,
rk+1(µ) := A(tk , µ)xk + f (xk , µ)− E (tk , µ)xk+1
= E (tk , µ)xk+1 − E (tk , µ)xk+1.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 17/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound
”Dual” system and the reduced ”dual” system
E (tk , µ)T xk+1du = −CT , W T
duE (tk , µ)TVduzk+1du = −W T
duCT .
Here, xkdu := Vduz
kdu approximates xk
du, k = 1, . . . ,Tn.
Residual of the reduced dual system:
rk+1du (µ) := −CT − E (tk , µ)T xk+1
du .
Recall residual of the ROM:
rk+1(µ) := A(tk , µ)xk + f (xk , µ)− E (tk , µ)xk+1.
Define an auxiliary vector,
rk+1(µ) := A(tk , µ)xk + f (xk , µ)− E (tk , µ)xk+1
= E (tk , µ)xk+1 − E (tk , µ)xk+1.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 17/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound (Cont.)
Theorem (Output Error Bound 2) [Zhang/Feng/Li/Benner ’15]
Assume that E (tk , µ) is invertible, then the output errorek
O(µ) := yk − yk satisfies
‖ekO(µ)‖ ≤ ∆k (µ), k = 1, . . . ,Tn,
where∆k (µ) := Φk (µ)||rk (µ)||,Φk (µ) = ‖E (tk−1, µ)−T‖‖rk
du(µ)‖+ ‖xkdu(µ)‖.
Define
ρk (µ) :=‖rk (µ)‖‖rk (µ)‖
.
It can be shown that ρk (µ) is bounded, i.e.,
ρk (µ) ≤ ρk (µ) ≤ ρk (µ).
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 18/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound (Cont.)
Theorem (Output Error Bound 2) [Zhang/Feng/Li/Benner ’15]
Assume that E (tk , µ) is invertible, then the output errorek
O(µ) := yk − yk satisfies
‖ekO(µ)‖ ≤ ∆k (µ), k = 1, . . . ,Tn,
where∆k (µ) := Φk (µ)||rk (µ)||,Φk (µ) = ‖E (tk−1, µ)−T‖‖rk
du(µ)‖+ ‖xkdu(µ)‖.
Define
ρk (µ) :=‖rk (µ)‖‖rk (µ)‖
.
It can be shown that ρk (µ) is bounded, i.e.,
ρk (µ) ≤ ρk (µ) ≤ ρk (µ).
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 18/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound (Cont.)Efficient Output Error Estimation: Case 1
Corollary 1 [Zhang/Feng/Li/Benner ’15]
Under the assumptions of Theorem 1, for all µ ∈ P, assume that
1 ‖rk (µ)‖: ∃α ∈ R+, s.t.,
α ≤ ‖rk+1(µ)‖/‖rk (µ)‖ ∀ k = 1, . . . ,Tn − 1;
2 f (·, µ) is Lipschitz continuous, i.e., ∃ Lf ∈ R+, s.t.,
‖f (x1, µ)− f (x2, µ)‖ ≤ Lf ‖x1 − x2‖, ∀ x1, x2 ∈ Wn;
3 Lf < α/‖E (tk , µ)−1‖.Then
ρk (µ) ≤ ρk (µ) ≤ ρk (µ),
where ρk (µ) = αα+Lf ‖(E(tk−2,µ)−1‖ , ρk (µ) = α
α−Lf ‖(E(tk−2,µ)−1‖ .
Remark: Assumption #3 is reasonable when ‖E (tk , µ)−1‖ . 1.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 19/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound (Cont.)Efficient Output Error Estimation: Case 2
Corollary 2 [Zhang/Feng/Li/Benner ’15]
Under the assumptions of Theorem 1, for all µ ∈ P, assume that
1 ‖rk (µ)‖: ∃α, α ∈ R+, s.t.,
α ≤ ‖rk (µ)‖/‖rk+1(µ)‖ ≤ α, ∀ k = 1, . . . ,Tn − 1;
2 f (·, µ) is bi-Lipschitz continuous, i.e., ∃ Lf , Lf ∈ R+, s.t.,
Lf ‖x1 − x2‖ ≤ ‖f (x1, µ)− f (x2, µ)‖ ≤ Lf ‖x1 − x2‖, x1, x2 ∈ Wn;
3 Lf > α−1/‖(E (tk , µ)−1‖.Then
ρk (µ) ≤ ρk (µ) ≤ ρk (µ),
where ρk (µ) = 1αLf ‖(E(tk−2,µ)−1‖+1
, ρk (µ) = 1α Lf ‖(E(tk−2,µ)−1‖−1
.
Remark: Assumption #3 is reasonable when ‖E (tk , µ)−1‖ is large.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 20/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound (Cont.)Efficient Output Error Estimation
Recall that ‖ekO(µ)‖ ≤ ∆k (µ) = Φk (µ)||rk (µ)||, ρk (µ) = ‖r k (µ)‖
‖r k (µ)‖ , we have:
Output Error Bound
‖ekO(µ)‖ ≤ ∆k (µ) := Φk (µ)ρk (µ)‖rk (µ)‖.
Estimating the Ratio ρk(µ)
ρk (µ) ≈ ρ? :=1
Tn
Tn∑k=1
ρk (µ?).
A computable output error estimation:
‖ekO‖ . ∆k
est(µ) := ρ?Φk (µ)‖rk (µ)‖.
Here, µ? is chosen to be the parameter, so that
µ? = argmaxµ∈P
ψ(µ), ψ(µ) =1
Tn
Tn∑k=1
∆kest(µ).
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 21/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Primal-dual Output Error Bound (Cont.)Efficient Output Error Estimation
Recall that ‖ekO(µ)‖ ≤ ∆k (µ) = Φk (µ)||rk (µ)||, ρk (µ) = ‖r k (µ)‖
‖r k (µ)‖ , we have:
Output Error Bound
‖ekO(µ)‖ ≤ ∆k (µ) := Φk (µ)ρk (µ)‖rk (µ)‖.
Estimating the Ratio ρk(µ)
ρk (µ) ≈ ρ? :=1
Tn
Tn∑k=1
ρk (µ?).
A computable output error estimation:
‖ekO‖ . ∆k
est(µ) := ρ?Φk (µ)‖rk (µ)‖.
Here, µ? is chosen to be the parameter, so that
µ? = argmaxµ∈P
ψ(µ), ψ(µ) =1
Tn
Tn∑k=1
∆kest(µ).
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 21/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
POD-Greedy Algorithm
How to compute V ?
Algorithm POD-Greedy [Haasdonk/Ohlberger ’08]
Input: Ptrain, µ0, εRB(< 1).Output: Reduced Basis (RB): V = [v1, . . . , vN ].
1: Initialization: N = 0, V = [ ], µ? = µ0, ψ(µ?) = 1.2: while ψ(µ?) > εRB do3: Compute the trajectory X := [x1(µ?), . . . , xTn (µ?)].4: POD process:
If N 6= 0, compute xk (µ?) := xk (µ?)−ProjW [xk (µ?)], k = 1, . . . ,Tn.Do SVD for X : X = QΣFT , vN+1 := Q(:, 1).Enrich V : V = [V , vN+1], W := colspanV .
5: N = N + 1.6: Find µ? := arg max
µ∈Ptrain
ψ(µ).
7: end while
Remark: When Tn is large, adaptive snapshot selection can be applied.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 22/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Adaptive Snapshot Selection (ASS)
The idea of ASS is to discard the redundant linear information in thetrajectory earlier, before the POD process.
x
θ
SA
• SA: selected snapshots subspace,
• x : to be tested,
• φ(SA, x): an indicator to measurethe linear dependency of SA and x , e.g.,
φ(SA, x) = ∠(SA, x).
• x is taken as a new snapshot only whenx is “sufficiently” linearly independentfrom SA, i.e., φ(SA, x) > εASS.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 23/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Adaptive Snapshot Selection (cont.)
Algorithm Adaptive Snapshot Selection [Zhang/Feng/Li/Benner ’14]
Input: xkTn
k=1, εASS.Output: Selected snapshot matrix SA = [xk1 , . . . , xk` ].
1: Initialization: j = 1, kj = 1, SA = [xkj ].2: for k = 2, . . . ,Tn do3: if φ
(SA, x
k)> εASS then
4: j = j + 1.5: kj = k .6: SA = [SA, x
kj ].7: end if8: end for
Remark: a relaxed condition φ(SA, xk ) = ∠(xkj , xk ) can be employed for
an efficient implementation.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 24/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Adaptive Snapshot Selection (cont.)
Algorithm Adaptive Snapshot Selection [Zhang/Feng/Li/Benner ’14]
Input: xkTn
k=1, εASS.Output: Selected snapshot matrix SA = [xk1 , . . . , xk` ].
1: Initialization: j = 1, kj = 1, SA = [xkj ].2: for k = 2, . . . ,Tn do3: if φ
(SA, x
k)> εASS then
4: j = j + 1.5: kj = k .6: SA = [SA, x
kj ].7: end if8: end for
Remark: a relaxed condition φ(SA, xk ) = ∠(xkj , xk ) can be employed for
an efficient implementation.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 24/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
ASS-POD-Greedy Algorithm
How to compute V ?
Algorithm POD-Greedy [Haasdonk/Ohlberger ’08]
Input: Ptrain, µ0, εRB(< 1).Output: Reduced Basis (RB): V = [v1, . . . , vN ].
1: Initialization: N = 0, V = [ ], µ? = µ0, ψ(µ?) = 1.2: while ψ(µ?) > εRB do3: Compute the trajectory X := [x1(µ?), . . . , xTn (µ?)].4: POD process:
If N 6= 0, compute xk (µ?) := xk (µ?)−ProjW [xk (µ?)], k = 1, . . . ,Tn.Do SVD for X : X = QΣFT , vN+1 := Q(:, 1).Enrich V : V = [V , vN+1], W := colspanV .
5: N = N + 1.6: Find µ? := arg max
µ∈Ptrain
ψ(µ).
7: end while
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 25/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
ASS-POD-Greedy Algorithm
POD-Greedy + ASS
Algorithm ASS-POD-Greedy [Zhang/Feng/Li/Benner ’14]
Input: Ptrain, εRB(< 1)Output: Reduced Basis (RB): V = [v1, . . . , vN ]
1: Initialization: N = 0, V = [ ], µ? = µ0, ψ(µ?) = 1.2: while ψ(µ?) > εRB do3: Compute the trajectory X := [x1(µ?), . . . , xTn (µ?)],
apply ASS to get: XASS := [xk1 (µ?), . . . , xk`(µ?)] (` Tn).4: POD process:
If N 6= 0, compute xkj (µ?) := xkj (µ?)− ProjW [xkj (µ?)], j = 1, . . . , `.Do SVD for XASS: XASS = QΣFT , vN+1 := Q(:, 1).Enrich V : V = [V , vN+1], W := colspanV .
5: N = N + 16: Find µ? := arg maxµ∈Ptrain ψ(µ).
7: end while
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 25/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Numerical Results
Numerical Examples:
1 Linear convection-diffusion equation
2 Burgers’ equation
3 Batch chromatography
4 Continuous SMB chromatography
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 26/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 1: Linear Convection-diffusion EquationPrimal-dual Error Bound/Estimation: Proposed vs. Existing
ut = q1uxx + q2ux − q2, x ∈ (0, 1), t ∈ (0, 1],
y =1
|Ω0|
∫Ω0
u(t, x) dx , Ω0 = [0.495, 0.505],
µ := (q1, q2), P = [0.1, 1]× [0.5, 5], n = 800, Tn = 100.
0 5 10 15 2010
−8
10−6
10−4
10−2
100
Size of RB: N
ma
x µ ∈
Ptr
ain
err
or
ErrorBound−1
ErrorBound−2
True error for ROM−1
True error for ROM−2ε
ROM=10
−6
Error bound decay during RB extension.ErrorBound-1: [Grepl/Patera’05], ErrorBound-2: proposed.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 27/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 1: Linear Convection-diffusion EquationBehavior of ρ?
0 5 10 15 200.5
1
1.5
2
2.5
3
Size of RB: N
Ra
tio
Behavior of the average ratio ρ? = 1Tn
Tn∑k=1
ρk (µ?) during the RB construction
process for the linear convection-diffusion equation.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 28/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 1: Linear Convection-diffusion EquationBehavior of the Ratio ‖rn+1‖/‖rn‖
0 20 40 60 80 1000
0.5
1
1.5
2
2.5
3
Time index: tn
Ra
tio
N=1
N=8
N=17
Behavior of the ratio ‖rn+1‖‖rn‖ in the time trajectory corresponding to different RB
dimensions for the linear convection-diffusion equation.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 29/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 2: Burgers’ EquationError Bound/Estimation: Primal Only vs. Primal-dual
ut + (u2
2)x = νuxx + 1, x ∈ (0, 1), t ∈ (0, 2],
y = u(t, 1; ν),
ν ∈ P = [0.05, 1], n = 500, Tn = 1000.
1 3 5 7 9 11 13
10−6
10−4
10−2
100
Size of RB: N
max
µ ∈
Ptr
ain
err
or
ErrorBound−1
ErrorBound−2
True errorε
ROM=10
−5
Error bound decay during RB extension.ErrorBound-1: primal only, ErrorBound-2: primal-dual.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 30/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 2: Burgers’ EquationBehavior of ρ?
1 3 5 7 9 11 132
4
6
8
10
12
14
Size of RB: N
Ra
tio
Behavior of the average ratio ρ? = 1Tn
Tn∑k=1
ρk (µ?) during the RB construction
process for the Burgers’ equation.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 31/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 2: Burgers’ EquationBehavior of the Ratio ‖rn+1‖/‖rn‖
0 200 400 600 800 10000
0.5
1
1.5
Time index: tn
Ra
tio
N=2
N=6
N=12
Behavior of the ratio ‖rn+1‖‖rn‖ in the time trajectory corresponding to different RB
dimensions for the Burgers’ equation.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 32/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 3: Batch Chromatography
Column
ba
Solvent
Pump
a + b
b
a
b a
t
int
cyct
Q 1t 2t 3t 4t
Feed
t
Products
concentration
concentration
Principle of batch chromatography for binary separation.
FOM ROM
Ack+1
z = Bckz + dk
z − τhkz
qk+1z = qk
z + ∆thkz
ckz , q
kz ∈ Rn, τ =
1− εε
∆t
Acza
k+1cz
= Bczakcz
+ dk0 dcz − τ Hczβ
kz
ak+1qz
= akqz
+ ∆tHqzβkz
akcz, ak
qz∈ RN , βk
z ∈ RM
n N,MThe parameter µ = (Q, tin).
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 33/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 3: Batch Chromatography
Column
ba
Solvent
Pump
a + b
b
a
b a
t
int
cyct
Q 1t 2t 3t 4t
Feed
t
Products
concentration
concentration
Principle of batch chromatography for binary separation.
FOM ROM
Ack+1
z = Bckz + dk
z − τhkz
qk+1z = qk
z + ∆thkz
ckz , q
kz ∈ Rn, τ =
1− εε
∆t
Acza
k+1cz
= Bczakcz
+ dk0 dcz − τ Hczβ
kz
ak+1qz
= akqz
+ ∆tHqzβkz
akcz, ak
qz∈ RN , βk
z ∈ RM
n N,MThe parameter µ = (Q, tin).
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 33/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 3: Batch ChromatographyPerformance of the ASS for Basis Generation
Illustration of the generation of CRBs (Wa, Wb) at the same error tolerance(εCRB = 1.0× 10−7) with different thresholds for ASS.
εASS Dim. CRB (Wa Wb) Runtime [h]no ASS – 146 152 62.5 (-)ASS 1.0× 10−4 147 152 6.05 (−90.3%)ASS 1.0× 10−3 147 152 3.62(−94.2%)ASS 1.0× 10−2 144 150 2.70 (−95.7%)
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 34/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 3: Batch ChromatographyPerformance of the ASS for Basis Generation
Comparison of the runtime for RB generation using the POD-Greedy algorithmwith and without ASS.
Algorithms Runtime [h]POD-Greedy 17.9ASS-POD-Greedy 7.6 (−57.5%)
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 35/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 3: Batch ChromatographyError Bound/Estimation: Primal Only vs. Primal-dual
6 11 16 21 26 31 36 41 46
10−5
10−3
10−1
101
103
Size of RB: N
max
µ ∈
Ptr
ain
error
ErrorBound−1
ErrorBound−2
True errorεROM
=1× 10−4
6
Ru
nti
me
6.8 h 7.6 h
ErrorBound-1 ErrorBound-2
Runtime for the RB construction.Error bound decay during RB extension.Size of RB: N
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 36/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 3: Batch ChromatographyBehavior of ρ?
5 15 25 35 450
5
10
15
20
25
30
35
40
45
Size of RB: N
Ra
tio
Behavior of the average ratio ρ? = 1Tn
Tn∑k=1
ρk (µ?) during the RB construction
process for the batch chromatographic model.
Size of RB: N
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 37/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 3: Batch ChromatographyROM-based Optimization
FOM-based Opt.:
minµ∈P−Pr(cz (µ), qz (µ);µ), s.t.
Rec(cz (µ), qz (µ);µ) ≥ Recmin,
cz (µ), qz (µ): solutions to FOM.
ROM-based Opt.:
minµ∈P−Pr(cz (µ), qz (µ);µ), s.t.
Rec(cz (µ), qz (µ);µ) ≥ Recmin,
cz (µ), qz (µ): solutions to ROM.
Approx.
Optimization based on the ROM (N = 45) and the FOM (n = 1500).
Model Obj. (Pr) Opt. solution (µ) #Iterations Runtime [h]/SpFFOM-Opt. 0.020264 (0.0796, 1.0545) 202 33.88 / -ROM-Opt. 0.020266 (0.0796, 1.0545) 202 0.58 / 58
? The optimizer: NLOPT GN DIRECT L in NLopt package.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 38/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 4: SMB Chromatography
SMB chromatographic process with 4 zones and 8 columns.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 39/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 4: SMB ChromatographyModel Descriptions
A more complex system:
1 More parameters: µ := (m1, . . . ,m4,QF ).
2 A multi-switching system: x0T +1 = Psx
Tn
T , T is the time period.
3 Cyclic steady state (CSS) computation, the system is simulatedmany time periods till the CSS is reached.
4 A parametric coupled system.
FOM:
Az (µ)ck+1
z = Bz (µ)ckz + rk
z + tsκzqkz
qk+1z = (1− tsκz ∆t)qk
z + tsκzHz ∆tckz
ROM:
Az (µ)ak+1
cz= Bz (µ)ak
cz+ rz + tsκz Dza
kqz
ak+1qz
= (1− tsκz ∆t)akqz
+ tsκzHz ∆tDTz ak
cz
Az (µ) = V TczAz (µ)Vcz , Bz (µ) = V T
czBz (µ)Vcz , rz = V T
czrkz , Dz = V T
czVqz .
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 40/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 4: SMB ChromatographyError Behavior during the RB Construction Process
1 21 41 61 81
10−4
10−3
10−2
10−1
100
Size of RB: N
max µ
∈ P
train
err
or
Error bound
True errorε
ROM= 10
−3
Error bound decay during RB extension.
Size of RB: N
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 41/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 4: SMB ChromatographyBehavior of ρ?
0 20 40 60 800
10
20
30
40
50
60
70
80
Size of RB: N
Ratio
Behavior of the average ratio ρ? = 1Tn
Tn∑k=1
ρk (µ?) during the RB construction
process for the SMB model.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 42/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 4: SMB ChromatographyROM Validation
Runtime comparison of the detailed and reduced simulations over a validationset Pval with 200 random sample parameters. εRB = 1× 10−3, εASS = 1× 10−5.
Simulations Maximal error Average runtime [s]/SpFFOM (n = 800) – 338.71(-)ROM (N = 83) 1.1× 10−4 46.7 / 7
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 43/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 4: SMB ChromatographyROM-based Optimization
FOM-based Opt.:
minµ∈P−QF(µ), s.t.,
Pua,E(cz (µ), qz (µ);µ) ≥ Pua,min,
Pub,R(cz (µ), qz (µ);µ) ≥ Pub,min,
Q1 ≤ Qmax,
cz (µ), qz (µ): solutions to FOM.
ROM-based Opt.:
minµ∈P−QF(µ), s.t.,
Pua,E(cz (µ), qz (µ);µ) ≥ Pua,min,
Pub,R(cz (µ), qz (µ);µ) ≥ Pub,min,
Q1 ≤ Qmax,
cz (µ), qz (µ): solutions to ROM.
Approx.
P = [4.2, 4.7]× [2.5, 3.0]× [3.5, 4.0]× [2.2, 2.7]× [0.05, 0.1],
Pua,E :=
∫ 1
0cE
a,CSS(t) dt∫ 1
0cE
a,CSS(t) dt +∫ 1
0cE
b,CSS(t)dt, Pub,R :=
∫ 1
0cR
b,CSS(t)dt∫ 1
0cR
a,CSS(t) dt +∫ 1
0cR
b,CSS(t) dt.
Constraints: Pua,min = 99.0%, Pub,min = 99.0%, Qmax = 0.50.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 44/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Example 4: SMB ChromatographyROM-based optimization
Comparison of the optimization based on the ROM (N = 83) and FOM(n = 800), εopt = 1× 10−4.
Initial-guess FOM-Opt. ROM-Opt.Objective QF 0.07 0.0745 0.0745
Opt. solution
m1 4.50 4.3269 4.3271m2 2.90 2.8599 2.8603m3 3.50 3.6036 3.6039m4 2.30 2.3468 2.3685QF 0.07 0.0745 0.0745
ConstraintsPua,E 98.89% 99.00% 99.00%Pub,R 99.49% 99.00% 99.00%Q1 0.4161 0.4997 0.4998
# Iterations 71 79Runtime [h] / SpF 5.13 / - 0.82 / 6
? The optimizer: NLOPT LN COBYLA in NLopt package.
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 45/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
Conclusions and Outlook
Conclusions:
An efficient output error estimation for MOR of nonlinearparametrized evolution equations is proposed.
Adaptive Snapshot Selection (ASS) is proposed, so that the offlinetime is largely reduced.
Application to convection dominated problems, e.g. batchchromatography and linear SMB chromatography, is presented.
Outlook:
More reliable and efficient estimation of ρk (µ).
Reduced basis methods for SMB chromatography with uncertaintyquantification (UQ).
Max Planck Institute Magdeburg P. Benner, MOR for Parameterized Evolution Equations 46/47
Motivation Model Order Reduction Error Bound Adaptive Snapshot Selection Numerical Examples Conclusions
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