Structure preserving reduced order modeling Jan S Hesthaven EPFL, - PowerPoint PPT Presentation

QUIET 2017 Structure preserving reduced order modeling Jan S Hesthaven EPFL, Lausanne, CH Jan.Hesthaven@epfl.ch w/ B. Maboudi, N. Ripamonti 
 EPFL, Lausanne, CH

QUIET 2017 Structure preserving reduced order modeling Jan S Hesthaven EPFL, Lausanne, CH Jan.Hesthaven@epfl.ch w/ B. Maboudi, N. Ripamonti 
 EPFL, Lausanne, CH

Model order reduction Let us consider ODE’s (or semi-discrete PDE’s) as ⇢ z ( µ ) t = L ( µ ) z ( µ ) + F ( µ, z ( µ )) z ( µ, 0) = z 0 ( µ ) where n � 1 z ∈ R n

Model order reduction Let us consider ODE’s (or semi-discrete PDE’s) as ⇢ z ( µ ) t = L ( µ ) z ( µ ) + F ( µ, z ( µ )) z ( µ, 0) = z 0 ( µ ) where n � 1 z ∈ R n Now we seek the reduced model z = A y A ∈ R n × k y ∈ R k n � k where

Model order reduction We seek a linear approximation to the solution manifold

Model order reduction By projection, we obtain the reduced system ⇢ A y ( µ ) t = L ( µ )A y ( µ ) + F ( µ, A y ( µ )) A y ( µ, 0) = A y 0 ( µ ) A + A = I and ⇢ y ( µ ) t = A + L ( µ )A y ( µ ) + A + F ( µ, A y ( µ )) y ( µ, 0) = y 0 ( µ )

Model order reduction By projection, we obtain the reduced system ⇢ A y ( µ ) t = L ( µ )A y ( µ ) + F ( µ, A y ( µ )) A y ( µ, 0) = A y 0 ( µ ) A + A = I and ⇢ y ( µ ) t = A + L ( µ )A y ( µ ) + A + F ( µ, A y ( µ )) y ( µ, 0) = y 0 ( µ ) Choosing the linear space - A - is clearly key Often done by accuracy ‣ POD ‣ Greedy approximation based on error

A known problem Consider the wave equation u tt − c 2 u xx = 0 Expressed as ⇢ q t = p p t = c 2 q xx Reduced model by POD

A known problem Consider the wave equation u tt − c 2 u xx = 0 Expressed as ⇢ q t = p p t = c 2 q xx Reduced model by POD

A known problem Consider shallow water equation ⇢ h t + r · ( h r φ ) = 0 2 | r φ | 2 + h = 0 φ t + 1 u = r φ

A known problem k=80 Consider shallow water equation ⇢ h t + r · ( h r φ ) = 0 2 | r φ | 2 + h = 0 φ t + 1 u = r φ Reduced model by POD

A known problem k=80 Consider shallow water equation ⇢ h t + r · ( h r φ ) = 0 k=160 2 | r φ | 2 + h = 0 φ t + 1 u = r φ Reduced model by POD

A known problem k=80 Consider shallow water equation ⇢ h t + r · ( h r φ ) = 0 k=160 2 | r φ | 2 + h = 0 φ t + 1 u = r φ Mode truncation instability Reduced model by POD

Problem ? Problem - we have destroyed delicate properties Systems are Hamiltonian

Problem ? Problem - we have destroyed delicate properties Systems are Hamiltonian Equations of evolution, 8 q = dH ˙ > > d p < p = � dH ˙ > > : d q ✓ q ◆ Or by defining y = p  � I n 0 y = J 2 n r y H ( y ) ˙ J 2 n = − I n 0

Problem ? Problem - we have destroyed delicate properties Systems are Hamiltonian Equations of evolution, 8 q = dH ˙ > > d p < p = � dH ˙ > > : d q ✓ q ◆ Or by defining y = p  � I n 0 y = J 2 n r y H ( y ) ˙ J 2 n = − I n 0 We must develop our reduced basis such that the reduced model maintains a Hamiltonian structure

Model order reduction Definition: A ∈ R 2 n × 2 k is a symplectic basis/transformation if: A T J 2 n A = J 2 k A T J 2 n A = J 2 k Definition: A set A of vectors A = { e 1 , . . . , e n } ∪ { f 1 , . . . , f n } is a symplectic basis if Ω ( e i , e j ) = Ω ( f i , f j ) = 0 , Ω ( f i , e j ) = δ i,j Ω ( v 1 , v 2 ) = v T 1 J 2 n v 2

Symplectic transformations ⇔ Symplectic Transformation: I A symplectic inverse of a symplectic matrix A is given by A + = J T 2 k A T J 2 k I If A is a symplectic matrix then (Peng et al. [2015]) I ( A + ) T is symplectic I A + A = I 2 k

Model order reduction Suppose for a symplectic subspace A 2 R 2 n × 2 k z ⇡ Ay, With substitution A ˙ y = J 2 n r z H ( Ay ) We require the residual be orthogonal to A : A + ( A ˙ y � J 2 n r z H ( Ay )) = 0 resulting r y ˜ ˜ y = A + J 2 n ( A + ) T ˙ H ( y ) , H ( y ) = H ( Ay ) | {z } J 2 k

Model order reduction Suppose for a symplectic subspace A 2 R 2 n × 2 k z ⇡ Ay, With substitution A ˙ y = J 2 n r z H ( Ay ) We require the residual be orthogonal to A : A + ( A ˙ y � J 2 n r z H ( Ay )) = 0 resulting r y ˜ ˜ y = A + J 2 n ( A + ) T ˙ H ( y ) , H ( y ) = H ( Ay ) | {z } J 2 k Since A is symplectic, reduced problem is symplectic

Reduced models Given set of Snapshots Y = [ y ( t 1 ) , . . . , y ( t N )] I Nonlinear optimization || Y − AA + Y || minimize A A T J 2 n A = J 2 k subject to I SVD based methods for basis generation. I Complex SVD, using q + i p . I Greedy approach.

Reduced models Given set of Snapshots Y = [ y ( t 1 ) , . . . , y ( t N )] I Nonlinear optimization || Y − AA + Y || minimize A A T J 2 n A = J 2 k subject to I SVD based methods for basis generation. I Complex SVD, using q + i p . I Greedy approach.

Reduced models Given set of Snapshots Y = [ y ( t 1 ) , . . . , y ( t N )] I Nonlinear optimization || Y − AA + Y || minimize A A T J 2 n A = J 2 k subject to I SVD based methods for basis generation. I Complex SVD, using q + i p . I Greedy approach. The Hamiltonian can be used as error estimator. H ( q , p ) = U ( q ) + K ( p ) = F 1 ( q , p ) + F 2 ( q , p )

The greedy method - error Energy Preservation Let ˆ z ( t ) := Ay ( t ) be the approximated solution. Energy loss associated with model reduction is ∆ H ( t ) := | H ( z ( t )) � H (ˆ z ( t )) | Now we have H (ˆ z ( t )) = H ( Ay ( t )) = ( H � A )( y ( t )) = ˜ H ( y ( t )) = ˜ H ( y 0 ) = ( H � A )( y 0 ) = H ( Ay 0 ) = H ( AA + z 0 ) meaning ∆ H ( t ) = | H ( z 0 ) � H ( AA + z 0 ) | , t � 0

The greedy method - algorithm The Greedy Algorithm Input: δ , Γ N = { ω 1 , . . . , ω N } , z 0 ( ω ) 1. ω ∗ ω 1 2. e 1 z 0 ( ω ∗ ) 3. f 1 J T 2 n z 0 ( ω ∗ ) 4. A [ e 1 , f 1 ] 5. while ∆ H ( ω ) > δ for all ω 2 Ω N w ∗ argmax 6. ∆ H ( ω ) ω ∈ Ω N 7. Compute trajectory snapshots S = { z ( t i , ω ∗ ) | i = 1 , . . . , M } z ∗ argmax k s � AA + s k 8. s ∈ S Apply symplectic Gram-Schmidt on z ∗ 9. e k +1 z ∗ / k z ∗ k 10. f k +1 J T 2 n z ∗ 11. 12. A [ e 1 , . . . , e k +1 , f 1 , . . . , f k +1 ] 13. end while

The greedy method - convergence Let S be a subset of R m and Y n , n  m , be a general n -dimensional subspace of R m . The Kolmogorov n -width of S in R m is given by d n ( S, R m ) := inf Y n sup y ∈ Y n k s � y k 2 inf s ∈ S Theorem Let S be a compact subset of R 2 n with exponentially small Kolmogorov n -width d k  c exp( � α k ) with α > log 3 . Then there exists β > 0 such that the symplectic subspaces A 2 k generated by the greedy algorithm provide exponential approximation properties such that k s � P 2 k ( s ) k 2  C exp( � β k ) for all s 2 S and some C > 0 .

Hamiltonian reduced model Wave equation: I size of original system : 1000 ( I size of reduced system : 30 q = p ˙ p = c 2 q xx ˙ I ∆ H = 5 × 10 − 4 . I || y − y r || L 2 = 5 × 10 − 5 Hamiltonian: H ( q, p ) = Z ✓ 1 ◆ 2 p 2 + 1 2 c 2 q 2 dx x Stability by construction

Hamiltonian reduced model Wave equation: I size of original system : 1000 ( I size of reduced system : 30 q = p ˙ p = c 2 q xx ˙ I ∆ H = 5 × 10 − 4 . I || y − y r || L 2 = 5 × 10 − 5 Hamiltonian: H ( q, p ) = Z ✓ 1 ◆ 2 p 2 + 1 2 c 2 q 2 dx x Stability by construction

Hamiltonian reduced model 10 3 POD 10 2 complex SVD cotangent lift 10 1 singularvalues 10 0 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 0 50 100 150 200 number of modes (d) (e) 10 5 POD 10 3 cotangent lift 10 1 complex SVD greedy 10 � 1 10 � 3 k e k L 2 10 � 5 10 � 7 10 � 9 10 � 11 10 � 13 10 � 15 0 5 10 15 20 25 30 t (f)

Symplectic Empirical Interpolation Nonlinear case: d ) d dt y = ˜ L + A + g ( A y ) dt z = L z + g ( z ) = Let H = H 1 + H 2 such that r z H 1 = L and r z H 2 = g . The (D)EIM DEIM approximation then is d L y + A + J 2 n U ( P T U ) − 1 P T g ( A y ) dt y = ˜ | {z } ˜ N ( y )

Symplectic Empirical Interpolation Nonlinear case: d ) d dt y = ˜ L + A + g ( A y ) dt z = L z + g ( z ) = Let H = H 1 + H 2 such that r z H 1 = L and r z H 2 = g . The (D)EIM DEIM approximation then is d L y + A + J 2 n U ( P T U ) − 1 P T g ( A y ) dt y = ˜ | {z } | {z } ˜ N ( y ) This system is a Hamiltonian system if and only if ˜ N ( y ) = J 2 k r y h ( y ) Note that g = r z H 2 = ( A + ) T r y H 2 . And if we take U = ( A + ) T N ( y ) = A + J 2 n ( A + ) T ( P T ( A + ) T ) − 1 P T ( A + ) T r y H 2 = J 2 k r y H 2 ( A y ) ˜

Schrödinger’s equation Schr¨ odinger Equation q t = p xx + ✏ ( q 2 + p 2 ) p, ( p t = − q xx − ✏ ( q 2 + p 2 ) q, With discrete Hamiltonian: N ✓ q i q i − 1 − q 2 + p i p i − 1 − p 2 ◆ + ✏ X i i 4( p 2 i + q 2 i ) 2 H ∆ x ( z ) = ∆ x ∆ x 2 ∆ x 2 i =1

Schrödinger’s equation 1 . 4 1 . 4 exact exact POD POD 1 . 2 cotangent lift 1 . 2 cotangent lift greedy greedy 1 . 0 1 . 0 0 . 8 0 . 8 | u | | u | 0 . 6 0 . 6 0 . 4 0 . 4 0 . 2 0 . 2 0 . 0 0 . 0 − 0 . 2 − 0 . 2 − 20 − 10 0 10 20 − 20 − 10 0 10 20 x x (g) t = 0 (h) t = 10 (i) t = 20