Adaptive rational Krylov subspaces for large-scale dynamical systems. V. Simoncini

Transcription

1 Adaptive rational Krylov subspaces for large-scale dynamical systems V. Simoncini Dipartimento di Matematica, Università di Bologna joint work with Vladimir Druskin, Schlumberger Doll Research 1

2 Given the continuous-time system Model Order Reduction Σ = A B C, A C n n Analyse the construction of a reduced system ˆΣ = à B C with à of size m n 2

3 Projection methods and Linear Dynamical Systems Time-invariant linear system: x (t) = Ax(t)+Bu(t), x(0) = x 0 y(t) = Cx(t) Approximation of the matrix Transfer function Solvers for the Lyapunov matrix equation Emphasis: A large dimensions, W(A) C 3

4 Projection methods: the general idea Given space K R n of size m and (orthonormal) basis V m, A A m = V mav m, B B m = V mb, C C m = CV m Solve with A m,b m,c m Choices of K in the literature: Standard Krylov subspace: K m (A,B) = span{b,ab,...,a m 1 B} Shift-Invert Krylov subspace: K m ((A σi) 1,B) = span{b,(a σi) 1 B,...,(A σi) (m 1) B}; often σ = 0 Extended Krylov subspace: EK m (A,B) = K m (A,B)+K m (A 1,A 1 B) Rational Krylov subspace: K m (A,B,s) = span{(a s 1 ) 1 B,(A s 2 ) 1 B,...,(A s m ) 1 B}; usually s = [s 1,...,s m ] a-priori 4

5 Projection methods: the general idea Given space K R n of size m and (orthonormal) basis V m, A A m = V mav m, B B m = V mb, C C m = CV m Solve with A m,b m,c m Choices of K in the literature: Standard Krylov subspace: K m (A,B) = span{b,ab,...,a m 1 B} Shift-Invert Krylov subspace: K m ((A σi) 1,B) = span{b,(a σi) 1 B,...,(A σi) (m 1) B}; often σ = 0 Extended Krylov subspace: EK m (A,B) = K m (A,B)+K m (A 1,A 1 B) Rational Krylov subspace: K m (A,B,s) = span{(a s 1 ) 1 B,(A s 2 ) 1 B,...,(A s m ) 1 B}; usually s = [s 1,...,s m ] a-priori 5

6 Projection methods: the general idea Given space K R n of size m and (orthonormal) basis V m, A A m = V mav m, B B m = V mb, C C m = CV m Solve with A m,b m,c m Choices of K in the literature: Standard Krylov subspace: K m (A,B) = span{b,ab,...,a m 1 B} Shift-Invert Krylov subspace: K m ((A σi) 1,B) = span{b,(a σi) 1 B,...,(A σi) (m 1) B}; often σ = 0 Extended Krylov subspace: EK m (A,B) = K m (A,B)+K m (A 1,A 1 B) Rational Krylov subspace: K m (A,B,s) = span{(a s 1 ) 1 B,(A s 2 ) 1 B,...,(A s m ) 1 B}; usually s = [s 1,...,s m ] a-priori 6

7 Projection methods: the general idea Given space K R n of size m and (orthonormal) basis V m, A A m = V mav m, B B m = V mb, C C m = CV m Solve with A m,b m,c m Choices of K in the literature: Standard Krylov subspace: K m (A,B) = span{b,ab,...,a m 1 B} Shift-Invert Krylov subspace: K m ((A σi) 1,B) = span{b,(a σi) 1 B,...,(A σi) (m 1) B}; often σ = 0 Extended Krylov subspace: EK m (A,B) = K m (A,B)+K m (A 1,A 1 B) Rational Krylov subspace: K m (A,B,s) = span{(a s 1 ) 1 B,(A s 2 ) 1 B,...,(A s m ) 1 B}; usually s = [s 1,...,s m ] a-priori 7

8 Projection methods: the general idea Given space K R n of size m and (orthonormal) basis V m, A A m = V mav m, B B m = V mb, C C m = CV m Solve with A m,b m,c m Choices of K in the literature: Standard Krylov subspace: K m (A,B) = span{b,ab,...,a m 1 B} Shift-Invert Krylov subspace: K m ((A σi) 1,B) = span{b,(a σi) 1 B,...,(A σi) (m 1) B}; often σ = 0 Extended Krylov subspace: EK m (A,B) = K m (A,B)+K m (A 1,A 1 B) Rational Krylov subspace: K m (A,B,s) = span{(a s 1 I) 1 B,(A s 2 I) 1 B,...,(A s m I) 1 B} usually s = [s 1,...,s m ] a-priori 8

9 Transfer function approximation h(ω) = c(a iωi) 1 b, ω [α,β] Given space K of size m and V s.t. K=range(V), h(ω) cv(v AV iωi) 1 (V b) Next: Classical benchmark experiment with Standard, Shift-invert and Extended Krylov 9

10 An example: CD Player, h(ω) = C i,: (A iωi) 1 B :,j 10 8 CDplayer, m=20, (1,1) 10 6 true Std Krylov 10 4 CDplayer, m=20, (1,2) 10 2 true Std Krylov CDplayer, m=20, (2,1) 10 1 true Std Krylov CDplayer, m=20, (2,2) 10 3 true Std Krylov

11 An example: CD Player, h(ω) = C i,: (A iωi) 1 B :,j 10 8 CDplayer, m=20, (1,1) 10 6 true Std Krylov 10 2 CDplayer, m=50, (1,2) 10 1 true Std Krylov CDplayer, m=20, (2,1) 10 1 true Std Krylov CDplayer, m=20, (2,2) 10 3 true Std Krylov

12 Rational Krylov Subspace Method. Choice of poles K m (A,B,s) = span{(a s 1 I) 1 B,(A s 2 I) 1 B,...,(A s m I) 1 B} cf. General discussion in Antoulas, Various attempts: Gallivan, Grimme, Van Dooren (1996, ad-hoc poles) Penzl ( , ADI shifts - preprocessing, Ritz values)... Sabino ( tuning within preprocessing) IRKA Gugercin, Antoulas, Beattie (2008) 12

13 A new adaptive choice of poles for RKSM K m (A,b,s) = span{(a s 1 I) 1 b,(a s 2 I) 1 b,...,(a s m I) 1 b} s = [s 1,...,s m ] to be chosen sequentially The fundamental idea: Assume you wish to solve (A si)x = b with a Galerkin procedure in K m (A,b,s). Let V m be orth. basis. The residual satisfies: b (A si)x m = r m(a)b m r m (s), r z λ j m(z) = z s j with λ j = eigs(v mav m ). Moreover, r m (A)b = min θ 1,...,θ m j=1 m (A θ j I)(A s j I) 1 b j=1 13

14 A new adaptive choice of poles for RKSM. Cont d r m (z) = m j=1 z λ j z s j, λ j = eigs(v mav m ) For A symmetric: ( s m+1 := arg max s [ λ max, λ min ] ) 1 r m (s) [λ min,λ max ] spec(a) (Druskin, Lieberman, Zaslavski (SISC 2010)) For A nonsymmetric: ( ) 1 s m+1 := arg max s S m r m (s) where S m C + approximately encloses the eigenvalues of A 14

15 Example of S m. CD Player, m = 12 S m : encloses mirrored current Ritz values: -eigs(v mav m ) and initial estimates: s (0) 1 = 0.1, s (0) 2,3 = 900±i x imaginary part real part poles + -eigs(a) S m 15

16 Transfer function evaluation. EADY Data set 10 3 true arnoldi invert arnoldi rational krylov h(ω) h(ω) h approx (ω) arnoldi invert arnoldi rational krylov frequency ω frequency ω m = 20. A of size 598 (nonsym), s (0) 1 = A /condest(a), s (0) 2,3 arg(maxr(eigs(a))) 16

17 Transfer function evaluation. Flow Data set h(ω) h(ω) h approx (ω) invert arnoldi rational krylov true invert arnoldi rational krylov m = 80. A,E of size 9669 (nonsym) s (0) 1 = A /(condest(a) E F ), s (0) 2 arg(max R(eigs(A))) (all real poles) 17

18 Comparison with optimal (a-priori) theoretical shifts. A sym Equidistributed nested sequence of real shifts (EDS) (from classical Zolotaryov sol n) asymptotically optimal rational space for ir Rate for the L error of h(ω): [ ( )] O exp π2 m(1+o(1)) 2log 4λ max λ min, for λ max λ min 1 Druskin, Knizhnerman, Zaslavsky (SISC 2009) 18

19 Comparison with optimal (a-priori) theoretical shifts. Cont d A R : Diagonal matrix with log-uniformly distributed values in [ 1, ] h(ω) h m (ω) h(ω) h approx (ω) (m = 20) arnoldi invert arnoldi rational krylov EDP poles max ω h(ω) h approx (ω) arnoldi invert arnoldi rational krylov EDS poles h(ω) h approx (ω) space dimension frequency range 19

20 The Lyapunov matrix equation AX +XA +bb = 0 Approximation by projection: K of dim. m, V m orthonormal basis. X X m = V m YV m V mav m Y +YV ma V m +V mbb V m = 0 Connection to Rational functions: X = R x(ω)x(ω) dω with x(ω) = (A ωii) 1 b 20

21 X X m = V m YV m with Approximation by projection (V mav m )Y +Y(V mav m ) +(V mb)(v mb) = 0 Some technical issues: K m (A,b,s) = span{b,(a s 2 I) 1 b,..., m j=2 (A s ji) 1 b} (includes b) All real poles (all real arithmetic work) Cheap computation of V mav m at each iteration m (K m (A,b,s) K m+1 (A,b,s)) Cheap computation of the residual norm R m = AX m +X m A +bb Cheap factorized form of solution X m = Xˆm := Zˆm Z ˆm, ˆm m 21

22 Some numerical experiments Competitors: ADI problem: computation of parameters Extended Krylov Subspace method outperforms ADI in general EK m (A,b) = K m (A,b)+K m (A 1,A 1 b) Comparison measures: Efficiency (CPU time) Memory (space dimension) Rank of solution 22

23 The RAIL (symmetric) data set n Rational Extended Rational Extended space space space space direct direct iterative iterative 1357 CPU time (s) dim. Approx. Space Rank of Solution CPU time (s) dim. Approx. Space Rank of Solution CPU time (s) dim. Approx. Space Rank of Solution Inner solves: PCG with IC(10 2 ) 23

24 The RAIL (symmetric) data set n Rational Extended Rational Extended space space space space direct direct iterative iterative 1357 CPU time (s) dim. Approx. Space Rank of Solution CPU time (s) dim. Approx. Space Rank of Solution CPU time (s) dim. Approx. Space Rank of Solution Inner solves: PCG with IC(10 2 ) 24

25 More Tests: two nonsymmetric problems n Rational Extended Rational Extended space space space space direct direct iterative iterative 9669 CPU time (s) dim. Approx. Space Rank of Solution CPU time (s) dim. Approx. Space Rank of Solution Convective thermal flow problems (FLOW, CHIP data sets) All real shifts used 25

26 Conclusions and outlook Adaptive procedure makes Rational Krylov subspace appealing Competitive in terms of both reduction space and CPU time Balanced Truncation? References: V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations SIAM J. Scient. Computing, v.29, n.3 (2007), pp V. Simoncini, The Extended Krylov subspace for parameter dependent systems Applied Num. Math. v.60 n.5 (2010) V. Druskin and V. Simoncini, Adaptive rational Krylov subspaces for large-scale dynamical systems pp.1-20, August (Under revision) 26