POD for Coupled Nonlinear PDE Systems Stefan Volkwein Department of - PowerPoint PPT Presentation

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References POD for Coupled Nonlinear PDE Systems Stefan Volkwein Department of Mathematics and Statistics, University of Constance Joined work with O. Lass and Stefan Trenz Int. Workshop on Control and Optimization, Graz 2011 Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References Motivation and Outline Multi component systems (battery equations) - PDEs with different types - nonlinear coupling → What is a good POD model? Optimization and model reduction - inexact second-order methods u ℓ δ � ≤ C � ζ ℓ � ¯ u δ − ¯ δ � - inexactness by model reduction → Can we ensure convergence (rate)? Nonlinear model reduction - nonlinear optimal control u ℓ � ≤ C � ζ ℓ � � ¯ u − ¯ - solve of reduced-order model → Can we apply error estimates? Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References POD-(D)EIM for coupled systems [Lass/V.’11] Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References Fine model (well-posedness see [Wu/Xu/Zou’06]) Elliptic-parabolic systems: T = 1, Ω = ( a , b ) y t − ∇ · ( c 1 ∇ y ) − N ( y , p , q ; µ ) = 0 in Q = (0 , T ) × Ω −∇ · ( c 2 ∇ p ) − N ( y , p , q ; µ ) = 0 in Q −∇ · ( c 3 ∇ q ) + N ( y , p , q ; µ ) = 0 in Q Parameter-dependent nonlinearity: µ = ( µ 1 , µ 2 ) ≥ 0 √ y sinh( µ 1 ( q − p − ln y )) N ( y , p , q ; µ ) = µ 2 Boundary conditions: y x ( t , a ) = y x ( t , b ) = p ( t , a ) = p x ( t , b ) = 0, q x ( t , a ) = q ( t , b ) = 0 Discretization: FE (2nd order) and implicit Euler method Numerical solution method: (damped) Newton algorithm Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References Reduced-Order Model (ROM) Fine model: y t − ∇ · ( c 1 ∇ y ) − N ( y , p , q ; µ ) = 0 in Q −∇ · ( c 2 ∇ p ) − N ( y , p , q ; µ ) = 0 (FM) in Q −∇ · ( c 3 ∇ q ) + N ( y , p , q ; µ ) = 0 in Q Idea of ROM: Replace (FM) by ROM, which is reliable (i.e., sufficiently accurate), but fast to evaluate Procedure: Galerkin projection of (FM) with appropriate ansatz function containing characteristics of (FM) Methods: Reduced-Basis, P roper O rthogonal D ecompostion,... Efficiency: decouple computation in off- and online phase, where the online phase is independent of discretization of (FM) Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References POD basis computation POD criterium: ℓ ≤ dim(span { y ( t ) | t ∈ [0 , T ] } ) � T ℓ � � 2 � � � min � y ( t ) − � y ( t ) , ψ i � ψ i s.t. � ψ i , ψ j � = δ ij d t � 0 i =1 Inner product: L 2 (Ω) or H 1 (Ω) (+b.c.) Solution to optimization problem: � T R ψ i = 0 � y ( t ) , ψ i � y ( t ) d t = λ i ψ i , i = 1 , . . . , ℓ � T ( K v i )( t ) = 0 � y ( t ) , y ( · ) � v i d s = λ i v i ( t ), i = 1 , . . . , ℓ 0 v i ( t ) y ( t ) d t / √ λ i � T Relation via SVD: ψ i = Discrete variant: α j = O (∆ t ) N t ℓ � � 2 � � � � � y ( t j ) − � y ( t j ) , ψ i � ψ i � ψ i , ψ j � = δ ij min α j s.t. � j =1 i =1 Solution: YY ⊤ ψ i = λ i ψ i , Y ⊤ Yv i = λ i v i , Yv i = √ λ i ψ i Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References ROM: different POD bases for y , p , and q Fine model: y t − div ( c 1 ∇ y ) − N ( y , p , q ; µ ) = 0 in Q (FM) − div ( c 2 ∇ p ) − N ( y , p , q ; µ ) = 0 in Q − div ( c 3 ∇ q ) + N ( y , p , q ; µ ) = 0 in Q FE model for (FM): y h ( t ) = � N FE i =1 ¯ y i ( t ) ϕ i etc. y ( t ) − ¯ M ¯ y t ( t ) + S c 1 ¯ N (¯ y ( t ) , ¯ p ( t ) , ¯ q ( t ); µ ) = 0 p ( t ) − ¯ S c 2 ¯ N (¯ y ( t ) , ¯ p ( t ) , ¯ q ( t ); µ ) = 0 q ( t ) + ¯ S c 3 ¯ N (¯ y ( t ) , ¯ p ( t ) , ¯ q ( t ); µ ) = 0 ROM for (FM): y ℓ ( t ) = � ℓ y y i ( t ) ψ y i =1 ˆ i etc. [Off-/Online] Ψ ⊤ y t ( t ) + Ψ ⊤ y ( t ) − Ψ ⊤ y ¯ N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) = 0 y M Ψ y ˆ y S c 1 Ψ y ˆ p ¯ Ψ ⊤ p ( t ) − Ψ ⊤ N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) = 0 p S c 2 Ψ p ˆ q ¯ Ψ ⊤ q ( t ) + Ψ ⊤ N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) = 0 q S c 3 Ψ q ˆ Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References Problems for the ROM ROM for (FM): y ℓ ( t ) = � ℓ y y i ( t ) ψ y i =1 ˆ i etc. M ℓ y ˆ y t ( t ) + S ℓ y y ¯ y ( t ) − Ψ ⊤ N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) = 0 c 1 ˆ S ℓ p p ¯ N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) = 0 p ( t ) − Ψ ⊤ c 2 Ψ p ˆ S ℓ q q ¯ q ( t ) + Ψ ⊤ N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) = 0 c 3 ˆ Problem 1: Imply the reconstruction error � T ℓ � � 2 � � � � � y ( t ) − � y ( t ) , ψ i � ψ i d t = λ i � 0 i =1 i >ℓ the error relation � y − y ℓ � 2 + � p − p ℓ � 2 + � q − q ℓ � 2 = O �� � i >ℓ λ i Problem 2: evaluation of the nonlinear terms y ¯ Ψ ⊤ N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) etc. is of complexity N FE ≫ ℓ Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References Problem 1: A-priori error estimation Problem 1: Imply the reconstruction error � T ℓ � � 2 � � � � � y ( t ) − � y ( t ) , ψ i � ψ i d t = λ i � 0 i =1 i >ℓ the error relation � y − y ℓ � 2 + � p − p ℓ � 2 + � q − q ℓ � 2 = O �� � i >ℓ λ i Theorem: There is a constant C > 0 such that � T 2 + � p ( t ) − p ℓ ( t ) � 2 + � q ( t ) − q ℓ ( t ) � 2 d t � y ( t ) − y ℓ ( t ) � 0 2 + �P ℓ y y t − y t � �P ℓ y y ◦ − y ℓ (0) � � 2 � ≤ C � � � � � λ y λ p λ q + C i + i + i i >ℓ y i >ℓ p i >ℓ q Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References y ¯ Problem 2: evaluation of Ψ ⊤ N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) Replacement: N ( y ℓ ( t ) , p ℓ ( t ) , q ℓ ( t ); µ ) ≈ � m F ( t ) = ¯ i =1 c i ( t ) u i ∈ R N FE . Interpolation condition for 1 ≤ k ≤ m ≪ N FE : m m � � � � � � � � F ( t ) p k = c ( t ) u i = c i ( t ) p k , p k ∈ { 1 , . . . , N FE } u i p k i =1 i =1 Computation of c ( t ): ( P T U ) c ( t ) = P T F ( t ) ∈ R m � �� � m × m Complexity reduction: P T F ( t ) = ¯ N ( P T y ℓ ( t ) , P T p ℓ ( t ) , P T q ℓ ( t ); µ ) Choice for U (DEIM): POD basis for span { F ( t j ) } N t j =0 . Theorem: error estimate for POD-DEIM [compare Chaturantabut/Sorensen] Alternative: EIM [Maday, Patera et al.] Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References Run 1: accuracy (a-priori analysis) for fixed parameter µ “Truth” solution: N x = 1000, N t = 100, 2 nd order elements POD and EIM: ℓ y = 12, ℓ p = 10, ℓ q = 10, ℓ DEIM = ℓ EIM = 25 Average relative L 2 error (FEM and POD): ROM ROM-EIM ROM-DEIM 1 . 6765 × 10 − 7 1 . 6763 × 10 − 7 1 . 6762 × 10 − 7 y 2 . 8723 × 10 − 7 2 . 7560 × 10 − 7 2 . 7467 × 10 − 7 p 9 . 7545 × 10 − 8 9 . 4332 × 10 − 8 9 . 1929 × 10 − 8 q CPU time: FEM POD EIM DEIM ROM ROM-EIM ROM-DEIM 18 . 20 0 . 20 0 . 19 0 . 03 6 . 03 0 . 24 ( ≈ 1/75) 0 . 48 Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References Run 2: multiple parameters Sample set: µ sample ∈ { 1 , 2 } × { 1 , 2 } Test set: µ test ∈ { 0 . 5 , 1 . 5 , 2 . 5 , 3 } × { 0 . 5 , 1 . 5 , 2 . 5 , 3 } POD and EIM: ℓ y = 20, ℓ p = 18, ℓ q = 18, ℓ EIM = ℓ DEIM = 40 CPU time: FEM POD EIM DEIM ROM ROM-EIM ROM-DEIM ∼ 18 0 . 54 0 . 74 0 . 09 ∼ 7 . 50 ∼ 0 . 30 ∼ 0 . 60 Average relative L 2 error: 2 error Average relative L 8 x 10 −7 (test set) ε Y ε P ε Q 6 ε 4 2 0 2 4 6 8 10 12 14 16 Parameter sample Stefan Volkwein POD for Coupled Nonlinear PDE Systems

Motivation & Outline POD-(D)EIM Multilevel SQP A-posteriori analysis Conclusion & References ROM based inexact/multilevel SQP [Kahlbacher/V.’11] Stefan Volkwein POD for Coupled Nonlinear PDE Systems