Model order reduction of large-scale dynamical systems with Jacobi-Davidson style eigensolvers

Transcription

1 MAX PLANCK INSTITUTE International Conference on Communications, Computing and Control Applications March 3-5, 2011, Hammamet, Tunisia. Model order reduction of large-scale dynamical systems with Jacobi-Davidson style eigensolvers P. Benner 1 M. E. Hochstenbach 2 P. Kürschner 1 1 Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory 2 Technische Universiteit Eindhoven Centre for Analysis, Scientific computing and Applications FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 1/15

2 Outline 1 Introduction and scientific background 2 Modal approximation 3 Dominant pole computation of large-scale systems 4 Current and future research, enhancements and generalizations Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 2/15

3 Linear time invariant systems Basic definitions Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) y(t) = C x(t) Coefficient matrices: system matrices A, E R n n, input matrix B R n m, output matrix C R p n. Involved functions: state / descriptor vector x(t) R n with initial condition x(t 0 ) = x 0, input / control u(t) R m, output y(t) R p. Some properties: n is the order of the system, E may be regular (state-space system) or singular (descriptor system). Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 3/15

4 Linear time invariant systems Applications and origins Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) Electrical engineering v1 v2 g(v) g(v) vξ 1 vξ g(v) g(v) y(t) = C x(t) i = u(t) g(v) C C C C C Numerical mechanics Biological and chemical networks, power systems,... Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 4/15

5 Linear time invariant systems Applications and origins Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) Electrical engineering v1 v2 g(v) g(v) vξ 1 vξ g(v) g(v) y(t) = C x(t) i = u(t) g(v) C C C C C Simulation Numerical mechanics Optimization Stabilization Biological and chemical networks, power systems,... Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 4/15

6 Linear time invariant systems Model order reduction Large-Scale LTI System E ẋ(t) = A x(t) + B u(t) y(t) = C x(t) Problem Modern applications lead to large system orders n, e.g., n 10 5 or higher. Computational costs, e.g. for simulation purposes, increase drastically. Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 5/15

7 Linear time invariant systems Model order reduction Large-Scale LTI System E ẋ(t) = A x(t) + B u(t) y(t) = C x(t) Model Order Reduction Solution Apply model order reduction to generate approximate system of low order k n with: Ã, Ẽ R k k, B R k m, C R p k, x(t) R k, u(t) R m, and ỹ(t) R p s.t. ỹ(t) y(t). Reduced Order Model Ẽ x(t) = Ã x(t) + B u(t) ỹ(t) = C x(t) Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 5/15

8 Modal approximation Eigenvalues and eigenvectors The right and left eigenvectors corresponding to a selected subset λ 1,..., λ k of eigenvalues of (A, E) satisfy the generalized eigenvalue problems: { Axj = λ j Ex j, x j 0 yj H A = λ j yj H ; j = 1,..., k. E, y j 0 Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 6/15

9 Modal approximation Eigenvalues and eigenvectors The right and left eigenvectors corresponding to a selected subset λ 1,..., λ k of eigenvalues of (A, E) satisfy the generalized eigenvalue problems: { Axj = λ j Ex j, x j 0 yj H A = λ j yj H ; j = 1,..., k. E, y j 0 Modal approximation uses X := [x 1,..., x k ], Y := [y 1,..., y k ] C n k as transformation matrices to get the reduced order model via: Y H E X x(t) = Y H A X x(t) + Y H B u(t); ỹ(t) = C X x(t) Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 6/15

10 Modal approximation Eigenvalues and eigenvectors The right and left eigenvectors corresponding to a selected subset λ 1,..., λ k of eigenvalues of (A, E) satisfy the generalized eigenvalue problems: { Axj = λ j Ex j, x j 0 yj H A = λ j yj H ; j = 1,..., k. E, y j 0 Modal approximation uses X := [x 1,..., x k ], Y := [y 1,..., y k ] C n k as transformation matrices to get the reduced order model via: Y H E X x(t) = Y H A X x(t) + Y H B u(t); ỹ(t) = C X x(t) Ẽ x(t) = Ã x(t) + B u(t); ỹ(t) = C x(t) Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 6/15

11 Modal approximation Dominant poles If (A, E) is regular and nondefective, the transfer function is given for s C by H(s) = C(sE A) 1 B = n f j=1 with residues R j := (Cx j )(yj H B) C p m. If R j s λ j + R R j 2 Re (λ j ) > R i 2 Re (λ i ), i j, i, j = 1,..., n f, then λ j is a dominant pole and (λ j, x j, y j ) a dominant eigentriplet. [Martins/Lima/Pinto 96, Rommes/Martins 06] Dominant poles significantly contribute to the system dynamics and cause peaks near Im (λ j ) in the frequency response plot. Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 7/15

12 Modal approximation Dominant poles Transfer function of artificial test system with E = I, blockdiagonal A and random B, C T R n. exact model, n = H(iω) ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 8/15

13 Modal approximation Dominant poles Transfer function of artificial test system with E = I, blockdiagonal A and random B, C T R n. Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 8/15

14 Modal approximation Dominant poles Transfer function of artificial test system with E = I, blockdiagonal A and random B, C T R n. 10 exact model, n = 217 Im (λ) of dominant poles H(iω) ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 8/15

15 Modal approximation Dominant poles Transfer function of artificial test system with E = I, blockdiagonal A and random B, C T R n. 10 exact model, n = 217 Im (λ) of dominant poles reduced, dom. poles, k = 21 H(iω) ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 8/15

16 Modal approximation Dominant poles Transfer function of artificial test system with E = I, blockdiagonal A and random B, C T R n. 10 exact model, n = 217 Im (λ) of dominant poles reduced, dom. poles, k = 21 reduced, smallest Re (λ), k = 21 H(iω) ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 8/15

17 Modal approximation Dominant poles Necessary Ingredient Transfer function of artificial test system with E = I, blockdiagonal A Algorithms and random B, for Ccomputing T R n. eigentriplets of large-scale, possibly nonnormal matrices are required. exact model, n = 217 Due to their cubic complexity andim memory (λ) of dominant requirements, poles direct methods (QR / QZ algorithm) reduced, are out dom. of the poles, game. k = reduced, smallest Re (λ), k = 21 One possible way out: Iterative projection based eigenvalue algorithms for the two-sided eigenvalue problem. H(iω) ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 8/15

18 Dominant pole computation of large-scale systems Two-sided eigenvalue methods General two-sided scheme Project (A, E) onto low-dimensional subspaces V = colsp(v ), V C n k, W = colsp(w ), W C n k, k n: ( ) W H ( A, E V = S, T ). Compute most dominant eigentriplet (θ, q, z) of (S, T ) and W H B, VC using methods for small matrices. Approximate dominant eigentriplet of (A, E) is (θ, v := Vq, w := Wz) with residuals r 1 := Av θev, r 2 := A T w θe T w. YES: eigentriplet (θ, v, w) converged! Are r 1, r 2 sufficiently small? NO: expand the subspaces V, W by some appropriate new basis vectors s, t and goto 1. Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 9/15

19 Dominant pole computation of large-scale systems Two-sided eigenvalue methods Different subspace expansions s, t lead to different methods. Two-sided Jacobi-Davidson, bi-e-orthogonal version [Stathopoulos 02, Hochstenbach/Sleijpen 03] Solve s, t approximately from the correction equations ( I Evw H ) (A θe) ( I vw H E ) s = r 1, ( I E T wv H) (A θe) H ( I wv H E T ) t = r 2. Dominant Pole Algorithm [Martins et al 96, Rommes et al 06/ 08] Solve s, t from (θe A)s = b, (θe A) H t = c. Rayleigh Quotient Iteration [Ostrowski 59, Parlett 74] Solve s, t from (A θe)s = Ev, (A θe) H t = E T w. Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 10/15

20 Dominant pole computation of large-scale systems Exact Solution Results for PEEC model of a patch antenna structure from the NICONET benchmark collection. 20 H(iω) 2 (db) exact n = 408 reduced k = ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 11/15

21 Dominant pole computation of large-scale systems Exact Solution Results for PEEC model of a patch antenna structure from the NICONET benchmark collection H(iω) H red (iω) 2 H(iω) 2 Relative error ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 11/15

22 Dominant pole computation of large-scale systems Iterative and inexact solution Assume (A, E) is really large-scale, such that exact solves with A θe are infeasible. Jacobi-Davidson style methods are known to be usually more robust with respect to inexact solves. [Voss 07] Apply, e.g., a limited number of steps of an iterative solver [GMRES, BiCG,...] to the correction equation. Requires appropriately projected preconditioner K p := ( I Evw H) K ( I vw H E ) with K A θe invertible. Application of K p given by: ( Kp 1 = I [K 1 (Ev)]w H ) E w H E[K 1 K 1. (Ev)] [Fokkema/Sleijpen/Van der Vorst 98, Rommes 07] Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 12/15

23 Dominant pole computation of large-scale systems Iterative and inexact solution preconditioning Often desired goal in large-scale eigenvalue computation: find eigenvalues close to τ C. use fixed preconditioner K A τe during whole iteration. Λ(A, E) target τ Im Re Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 13/15

24 Dominant pole computation of large-scale systems Iterative and inexact solution preconditioning Often desired goal in large-scale eigenvalue computation: find eigenvalues close to τ C. use fixed preconditioner K A τe during whole iteration. Dominant poles are often scattered in C. variable preconditioner K A θe might be required. Λ(A, E) target τ Λ(A, E) dom. poles Im Im Re Re Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 13/15

25 Dominant pole computation of large-scale systems Iterative and inexact solution Results for FDM model of semi-discretized heat equation with convection-reaction taken from LyaPack. Approximate solution of correction equation using GMRES until r GMRES < 10 3 or 25 steps are processed. Preconditioner: incomplete LU-factorization K = LŨ A θi with drop tolerance If r GMRES > 10 3 after 25 2 = 13 steps, K is updated for the next iteration of two-sided JD. Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 14/15

26 Dominant pole computation of large-scale systems Iterative and inexact solution Results for FDM model of semi-discretized heat equation with convection-reaction taken from LyaPack. 40 H(iω) 2 (db) exact n = reduced k = ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 14/15

27 Dominant pole computation of large-scale systems Iterative and inexact solution Results for FDM model of semi-discretized heat equation with convection-reaction taken from LyaPack H(iω) H red (iω) 2 H(iω) 2 Relative error ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 14/15

28 Current and future research, enhancements and generalizations Efficient preconditioning: Which kind of preconditioner (esp. difficult if E is singular)? When should K be renewed / updated? Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 15/15

29 Current and future research, enhancements and generalizations Efficient preconditioning. Robust handling of occurring oblique projectors I Evw H. Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 15/15

30 Current and future research, enhancements and generalizations Efficient preconditioning. Robust handling of occurring oblique projectors I Evw H. Generalizations to second order systems Mẍ(t) + Dẋ(t) + Kx(t) = Bu(t), y(t) = Cx(t). Exact solves Quadratic DPA [Martins/Rommes 08] Inexact solves Quadratic JD [Booten et al 96, Hochstenbach/Sleijpen 03] Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 15/15

31 Current and future research, enhancements and generalizations Efficient preconditioning. Robust handling of occurring oblique projectors I Evw H. Generalizations to second order systems. Combinations of modal approximation with other MOR approaches: dominant poles as (additional) interpolation points for Krylov subspace methods : j max { X = K q (A σj E) 1 E, (A σ j E) 1 B }, j=1 j max { Y = K q (A σj E) H E T, (A σ j E) H C T }. j=1 [Grimme 97, Rommes 07] Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 15/15

32 Current and future research, enhancements and generalizations Efficient preconditioning. Robust handling of occurring oblique projectors I Evw H. Generalizations to second order systems. Combinations of modal approximation with other MOR approaches: dominant poles as (additional) interpolation points for Krylov subspace methods. dominant poles as (additional) shift parameters for ADI method used to solve Lyapunov equations for balanced truncation: Re (p i ) ( Z j = I (pj + p j 1 )(A + p j E) 1 E ) Zj 1 Re (p i 1 ) [See our talk at MODRED 2010, Dec. 2-4, Berlin, available at Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 15/15

33 Current and future research, enhancements and generalizations Efficient preconditioning. Robust handling of occurring oblique projectors I Evw H. Generalizations to second order systems. Combinations of modal approximation with other MOR approaches. H(iω) original, n = BT, k = 160, (ritz values) BT, k = 160, (ritz + d.p.) ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 15/15

34 Current and future research, enhancements and generalizations Efficient preconditioning. Robust handling of occurring oblique projectors I Evw H. Generalizations to second order systems. Combinations of modal approximation with other MOR approaches. relative error BT, k = 160, (ritz values) BT, k = 160, (ritz + d.p.) ω Max Planck Institute Magdeburg P. Kürschner, MOR with Jacobi-Davidson 15/15