The Loewner Framework for Model Reduction of Flow Equations - PowerPoint PPT Presentation

The Loewner Framework for Model Reduction of Flow Equations Matthias Heinkenschloss Department of Computational and Applied Mathematics Rice University, Houston, Texas heinken@rice.edu February 18, 2020 Workshop Mathematics of Reduced Order Models ICERM, Brown University, February 17 - 21, 2020 Joint work with A. C. Antoulas (ECE, Rice U. & MPI Magdeburg) and I. V. Gosea (MPI Magdeburg) Funded in part by NSF grants CCF-1816219 and DMS-1819144 Matthias Heinkenschloss Feb. 18, 2020 1

Motivation ◮ Full order model (FOM) (typically discretization of system of PDEs) E d dt y ( t ) = Ay ( t ) + F ( y ( t )) + Bu ( t ) with state y ( t ) ∈ R n , n ≫ 1 , input u ( t ) ∈ R m , and output z ( t ) = Cy ( t ) + Du ( t ) ∈ R p . ◮ Projection based reduced order model (ROM): Matrices V , W ∈ R n × r , r ≪ n ; ROM W T EV d y ( t ) = W T AV � y ( t ) + W T F ( V � y ( t )) + W T Bu ( t ) dt � y ( t ) ∈ R r , r ≪ n , input u ( t ) ∈ R m , and output with state � y ( t ) + Du ( t ) ∈ R p . � z ( t ) = CV � ◮ Goal: Input-to-output map u �→ � z of ROM approximates input-to-output map u �→ z of FOM. ◮ Notation: States y , input u , output z . Matthias Heinkenschloss Feb. 18, 2020 2

Many ROM methods, incl. ◮ Proper Orthogonal Decomposition (POD) (e.g., books and survey articles [Gubisch and Volkwein, 2017], [Hesthaven et al., 2015], [Quarteroni et al., 2016]). ◮ Reduced Basis (RB) methods (e.g., books and survey articles [Haasdonk, 2017], [Hesthaven et al., 2015], [Quarteroni et al., 2016], [Rozza et al., 2008]). ◮ Balanced Truncation and Balanced POD (e.g., book [Antoulas, 2005] and survey articles [Benner and Breiten, 2017], [Rowley and Dawson, 2017]). ◮ Rational Interpolation (e.g., book [Antoulas et al., 2020a] and survey article [Beattie and Gugercin, 2017]). All require access to E , A , . . . to generate projection matrices E = W T EV , � A = W T AV , . . . V , W ∈ R n × r , then generate ROM � This talk: Loewner framework ◮ Constructs ROM � E , � A , . . . directly from data. ◮ For LTI systems [Antoulas et al., 2020a], [Antoulas et al., 2017]; recent work on descriptor systems, incl. Oseen equations (e.g., [Antoulas et al., 2020b], [Gosea et al., 2020]). ◮ Extension to quadratic-bilinear systems [Gosea and Antoulas, 2018], Burger’s equation [Antoulas et al., 2018]. Matthias Heinkenschloss Feb. 18, 2020 3

Loewner for Linear Time-Invariant (LTI) Systems ◮ Consider linear time-invariant system E d dt y ( t ) = Ay ( t ) + b u ( t ) with state y ( t ) ∈ R n , n ≫ 1 , input u ( t ) ∈ R , and output z ( t ) = c T y ( t ) + d u ( t ) ∈ R . Example: Oseen equations. ◮ To simplify presentation consider case of single input b ∈ R n , u ( t ) ∈ R , and single output c ∈ R n , z ( t ) ∈ R (SISO). Multiple inputs, multiple outputs (MIMO) can be handled via so-called left and right tangential directions, but is a bit more technical and requires more notation. Matthias Heinkenschloss Feb. 18, 2020 4

Frequency Domain ◮ Frequency domain s ∈ i R s E y ( s ) = Ay ( s ) + b u ( s ) , z ( s ) = c T y ( s ) . From now on we work in frequency domain; use same notation y , u , z for variables in frequency domain. ◮ Transfer function H ( s ) = c T ( s E − A ) − 1 b . E = W T EV , � A = W T AV , � b = W T b , � c = V T c , ◮ Seek ROM � s � y ( s ) = � y ( s ) + � E � A � b u ( s ) , c T � � z ( s ) = � y ( s ) + d u ( s ) , so that transfer function c T ( s � b ≈ H ( s ) = c T ( s E − A ) − 1 b . � E − � A ) − 1 � H ( s ) = � Matthias Heinkenschloss Feb. 18, 2020 5

Interpolation ◮ Given distinct frequencies µ 1 , . . . µ r , λ 1 , . . . λ r ∈ C , want ROM s.t. c T ( µ j � H ( µ j ) = H ( µ j ) = c T ( µ j E − A ) − 1 b , j = 1 , . . . r, E − � A ) − 1 � b = � � c T ( λ j � H ( λ j ) = H ( λ j ) = c T ( λ j E − A ) − 1 b , j = 1 , . . . r. E − � A ) − 1 � b = � � ◮ Theorem: Interpolation guaranteed if ROM projection matrices V , W satisfy ( λ j E − A ) − 1 b ∈ R ( V ) , j = 1 , . . . r, � c T ( µ j E − A ) − 1 � T ∈ R ( W ) , j = 1 , . . . r. ◮ Can choose (assume these have full rank) � � ( λ 1 E − A ) − 1 b , . . . , ( λ r E − A ) − 1 b ∈ C n × r , V =   c T ( µ 1 E − A ) − 1  .  W ∗ =  ∈ C r × n . .  . c T ( µ r E − A ) − 1 Matthias Heinkenschloss Feb. 18, 2020 6

Elementary identities Recall H ( s ) = c T ( s E − A ) − 1 b . c T ( µ E − A ) − 1 E ( λ E − A ) − 1 b � � 1 c T ( µ E − A ) − 1 ( µ − λ ) E ( λ E − A ) − 1 b = µ − λ � c T ( µ E − A ) − 1 � � � 1 ( λ E − A ) − 1 b = ( µ E − A ) − ( λ E − A ) µ − λ � � 1 = H ( λ ) − H ( µ ) . µ − λ c T ( µ E − A ) − 1 A ( λ E − A ) − 1 b = − c T ( µ E − A ) − 1 ( µ E − A ) ( λ E − A ) − 1 b + µ c T ( µ E − A ) − 1 E ( λ E − A ) − 1 b = − H ( λ ) + µ c T ( µ E − A ) − 1 E ( λ E − A ) − 1 b � � 1 = − H ( λ ) + µ H ( λ ) − H ( µ ) µ − λ � � 1 = λ H ( λ ) − µ H ( µ ) . µ − λ Matthias Heinkenschloss Feb. 18, 2020 7

� � ( λ 1 E − A ) − 1 b , . . . , ( λ r E − A ) − 1 b ∈ C n × r , If V =   c T ( µ 1 E − A ) − 1   . W ∗ =  ∈ C r × n , .  . c T ( µ r E − A ) − 1 (recall H ( s ) = c T ( s E − A ) − 1 b ) then   H ( µ 1 ) − H ( λ 1 ) H ( µ 1 ) − H ( λ r ) · · · µ 1 − λ 1 µ 1 − λ r   . . ... �   E = W ∗ EV = − def . . = − L , . .   H ( µ r ) − H ( λ 1 ) H ( µ r ) − H ( λ r ) · · · µ r − λ 1 µ r − λ r   µ 1 H ( µ 1 ) − H ( λ 1 ) λ 1 µ 1 H ( µ 1 ) − H ( λ r ) λ r · · · µ 1 − λ 1 µ 1 − λ r   . . ... �   A = W ∗ AV = − def . . = − L s , . .   µ r H ( µ r ) − H ( λ 1 ) λ 1 µ r H ( µ r ) − H ( λ r ) λ r · · · µ r − λ 1 µ r − λ r b = W ∗ b = [ H ( µ 1 ) , . . . , H ( µ r )] T , c = V ∗ c = [ H ( λ 1 ) , . . . , H ( λ r )] T . � � Matthias Heinkenschloss Feb. 18, 2020 8

� � ( λ 1 E − A ) − 1 b , . . . , ( λ r E − A ) − 1 b ∈ C n × r , If V =   c T ( µ 1 E − A ) − 1   . W ∗ =  ∈ C r × n , .  . c T ( µ r E − A ) − 1 (recall H ( s ) = c T ( s E − A ) − 1 b ) then   H ( µ 1 ) − H ( λ 1 ) H ( µ 1 ) − H ( λ r ) · · · µ 1 − λ 1 µ 1 − λ r   . . ... �   E = W ∗ EV = − def . . = − L , . .   H ( µ r ) − H ( λ 1 ) H ( µ r ) − H ( λ r ) · · · µ r − λ 1 µ r − λ r   µ 1 H ( µ 1 ) − H ( λ 1 ) λ 1 µ 1 H ( µ 1 ) − H ( λ r ) λ r · · · µ 1 − λ 1 µ 1 − λ r   . . ... �   A = W ∗ AV = − def . . = − L s , . .   µ r H ( µ r ) − H ( λ 1 ) λ 1 µ r H ( µ r ) − H ( λ r ) λ r · · · µ r − λ 1 µ r − λ r b = W ∗ b = [ H ( µ 1 ) , . . . , H ( µ r )] T , c = V ∗ c = [ H ( λ 1 ) , . . . , H ( λ r )] T . � � Only need data ( µ j , H ( µ j )) , ( λ j , H ( λ j )) , j = 1 , . . . , r , to construct ROM � E , � A , � b , � c ! Matthias Heinkenschloss Feb. 18, 2020 8

Loewner Framework ◮ Given distinct frequencies µ 1 , . . . µ k , λ 1 , . . . λ k ∈ C and transfer function measurements H ( µ 1 ) , . . . , H ( µ k ) , H ( λ 1 ) , . . . , H ( λ k ) ∈ C , ◮ use Loewner matrix   H ( µ 1 ) − H ( λ 1 ) H ( µ 1 ) − H ( λ k ) · · · µ 1 − λ 1 µ 1 − λ k   . . ...   . .  ∈ C k × k L = . .  H ( µ k ) − H ( λ 1 ) H ( µ k ) − H ( λ k ) · · · µ k − λ 1 µ k − λ k and shifted Loewner matrix   µ 1 H ( µ 1 ) − H ( λ 1 ) λ 1 µ 1 H ( µ 1 ) − H ( λ k ) λ k · · · µ 1 − λ 1 µ 1 − λ k   . . ...   ∈ C k × k  . . L s = . .  µ k H ( µ k ) − H ( λ 1 ) λ 1 µ k H ( µ k ) − H ( λ k ) λ k · · · µ k − λ 1 µ k − λ k ◮ to directly compute ROM � E , � A ∈ R r × r , � c ∈ R r , such that b , � c T ( µ j � H ( µ j ) ≈ H ( µ j ) = c T ( µ j E − A ) − 1 b , j = 1 , . . . k, A ) − 1 � E − � b = � � c T ( λ j � H ( λ j ) ≈ H ( λ j ) = c T ( λ j E − A ) − 1 b , j = 1 , . . . k. E − � A ) − 1 � b = � � Matthias Heinkenschloss Feb. 18, 2020 9