Structured Krylov Subspace Methods for Eigenproblems with Spectral Symmetries

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1 Structured Krylov Subspace Methods for Eigenproblems with Spectral Symmetries Fakultät für Mathematik TU Chemnitz, Germany Peter Benner joint work with Heike Faßbender (TU Braunschweig) and Hongguo Xu (University of Kansas) Theoretical and Computational Aspects of Matrix Algorithms Dagstuhl, October 12 17, 2003

2 Overview Overview The Hamiltonian eigenproblem Applications The symplectic Lanczos method Choosing the free parameters Numerical examples Conclusions Peter Benner TU Chemnitz 2

3 Hamiltonian eigenproblem The Hamiltonian Eigenproblem Hx = λx H R 2n 2n is a Hamiltonian matrix, i.e., (HJ) T = HJ, where J = [ 0 Hamiltonian matrices have explicit block structure implied by symmetry of HJ, H = [ A B C A T ] I n I n 0, B = B T, C = C T ; are skew-selfadjoint w.r.t. indefinite inner product x, y J := x T Jy. form Lie algebra with corresponding Lie group: S n := {S R 2n 2n SJS T = J} = symplectic matrices, hence S 1 HS is Hamiltonian for all S S 2n. ]. Peter Benner TU Chemnitz 3

4 Hamiltonian eigenproblem Spectral Properties Spectral symmetry with respect to real and imaginary axes: λ Λ (H) λ, ±λ Λ (H) Hence, λ R ±λ Λ (H), λ = jω jr ±jω Λ (H), λ C \ jr ±λ, ±λ Λ (H), Peter Benner TU Chemnitz 4

5 applications Systems and control: Applications Solution methods for algebraic and differential Riccati equations. Design of LQR/LQG/H 2 /H controllers and filters for continuous-time linear control systems. Stability radii and system norm computations; optimization of system norms. Passivity-preserving model reduction based on balancing. Reduced-order control for infinite-dim. systems based on inertial manifolds. Computational physics: exponential inegrators for Hamiltonian dynamics. Quantum chemistry: computing excitation energies in many-particle systems using random phase approximation (RPA). Quadratic eigenvalue problems... Peter Benner TU Chemnitz 5

6 applications Quadratic Eigenvalue Problems Consider λ 2 Mx + λgx + Kx = 0, where M = M T spd, K = K T, G = G T. Motivation: linear elasticity: computation of corner singularities in 3D anisotropic elastic structures [Apel/Mehrmann/Watkins 01] FE discretization in structural analysis [H.R. Schwarz 84] acoustic simulation of poro-elastic materials [Meerbergen 99] gyroscopic systems, rotor systems [Lancaster 99] Peter Benner TU Chemnitz 6

7 applications Spectral Symmetry of QEP Spectrum of matrix polynomial λ 2 Mx + λgx + Kx = 0, G = G T, looks like Hamiltonian symmetry! Peter Benner TU Chemnitz 7

8 applications Linearization of QEP Setting y = λx yields linear (generalized) eigenvalue problem ([ 0 K M 0 ] λ [ M G 0 M ]) [ y x ] = 0. H = N = [ ] 0 K M 0 [ ] M G 0 M is Hamiltonian: (HJ) T = HJ is skew-hamiltonian: (NJ) T = NJ = H λn is a Hamiltonian/skew-Hamiltonian pencil [Mehrmann/Watkins 00] Peter Benner TU Chemnitz 8

9 applications Re-formulation of H λn Note: N = Z 1 Z 2 = [ I 1 2 G 0 M ] [ M 1 2 G 0 I ]. Hence, as M is spd where Z1 1 (H λn)z 1 2 = W λi, W = [ I 1 2 G 0 I ] [ 0 K M 1 0 ] [ I 1 2 G 0 I ] W is Hamiltonian! Peter Benner TU Chemnitz 9

10 symplectic Lanczos method Observations: Structured Lanczos Method for Hamiltonian Matrices Preservation of spectral symmetry requires structured eigensolver. Hamiltonian structure is preserved under symplectic similarity transformations. For large, sparse problems want Krylov subspace method. = Need symplectic basis for Krylov subspace K(H, 2k, v) := span{v, Hv, H 2 v,..., H 2k 1 }. H nonsymmetric apply Arnoldi method or nonsymmetric Lanczos method. Arnoldi method can only produce symplectic Krylov basis in very particular instances! [Ammar/Mehrmann 91] = Need variant of nonsymmetric Lanczos! Peter Benner TU Chemnitz 10

11 symplectic Lanczos method Symplectic Lanczos Transformation For every Hamiltonian matrix H there exists symplectic similarity transformation H = S 1 HS = δ 1 β 1 ζ 2 δ 2 ζ 2 β 2 ζ 3 δ 3 ζ ζ n δ n ζ n β n ν 1 δ 1 ν 2 δ 2 ν 3 δ ν n δ n Hamiltonian J-tridiagonal (J-Hessenberg) form = Need 4n 1 parameters to represent H. Peter Benner TU Chemnitz 11

12 symplectic Lanczos method Relation of Reduced Form and Krylov Subspace Theorem: If H = S 1 HS is in Hamiltonian J-tridiagonal form, then K(H, 2n 1, v) = SR with s 1 = v is an SR decomposition of the Krylov matrix K(H, 2n 1, v) := [ v, Hv,..., H 2n 1 v ]. Peter Benner TU Chemnitz 12

13 symplectic Lanczos method Some Notation Permutation using P := [e 1, e 3,..., e 2n 1, e 2, e 4,..., e 2n ] (perfect shuffle) yields H P := P HP T, S P := P SP T, J P := P JP T = diag ([ ],..., [ ]) H P := S 1 P H P S P = δ 1 β 1 0 ζ 2 ν 1 δ ζ 2 δ 2 β 2 0 ζ ν 2 δ ζ ζ n ζ n δ n β n 0 0 ν n δ n Peter Benner TU Chemnitz 13

14 symplectic Lanczos method Derivation of Symplectic Lanczos Algorithm via J-Tridiagonalization Compute (permuted) J-tridiagonal form of a (permuted) Hamiltonian matrix H (H P ) by column-wise evaluation of HS = S H (H P S P = S P HP, where S P = [v 1, w 1, v 2, w 2,..., v n, w n ] is a permuted symplectic matrix). = H P v m = δ m v m + ν m w m ν m w m = H P v m δ m v m =: w m H P w m = ζ m v m 1 + β m v m δ m w m + ζ m+1 v m+1 ζ m+1 v m+1 = H P w m ζ m v m 1 β m v m + δ m w m =: ṽ m+1. = Choose parameters δ m, β m, ν m, ζ m such that resulting algorithm computes J P -orthogonal basis of Krylov subspace K(H P, v 1, l) = span{v 1, H P v 1,..., H 2l 1 P v 1 }. Peter Benner TU Chemnitz 14

15 symplectic Lanczos method Want S T P J P S P = J P = Choice of Parameters ν m+1 = vm+1j T P H P v m+1, β m = wmj T P H P w m,. ζ m 0, δ m are free! ζ m = ṽ m 2 = v m 2 = 1 Possible choices for δ m : δ m = 1 [B./Faßbender 97] δ m = v T mh P v m = v m w m [Lin/Ferng/Wang 97] δ m = 0 = algorithm is equivalent to HR/HZ tridiagonalization [B./Faßbender/Watkins 97, Watkins 02] Peter Benner TU Chemnitz 15

16 symplectic Lanczos method Generic Symplectic Lanczos Algorithm Choose initial vector ṽ 1 R 2n, ṽ 1 0. m = 1; v 0 = 0; ζ 1 = ṽ 1 2 ; ν 0 = 1 while ζ m 0 and ν m 1 0 v m = ṽ m /ζ m w m = H P v m δ m v m ν m = v T mj P H P v m if ν m 0 then w m = w m /ν m β m = w T mj P H P w m ṽ m+1 = H P w m ζ m v m 1 β m v m + δ m w m ζ m+1 = ṽ m+1 2 end if m = m + 1 end while Peter Benner TU Chemnitz 16

17 symplectic Lanczos method Lanczos Recursion and Breakdown H 2k P Define SP 2k = [v 1, w 1, v 2, w 2,..., v k, w k ] R 2n 2k, let principal submatrix of H P, then the symplectic Lanczos recursion is be the leading 2k 2k H P S 2k P = S 2k P 2k M P + ζ k+1 v k+1 e T 2k. Breakdown is possible: ṽ m = 0 ζ m = 0, lucky breakdown! w m = 0 ν m = 0, lucky breakdown! symplectic invariant subspace of H is detected, invariant subspace of H P of dimension 2m 1 is detected, ν m = 0 but v m 0 and w m 0 = serious breakdown! Peter Benner TU Chemnitz 17

18 choosing the free parameters Theorem: Uniqueness of Lanczos Vectors If there are two symplectic Lanczos recurrences with v 1 = v 1 HV k = V k H k + ζ k+1 ṽ k+1 e T 2k, HV k = V k H k + ζ k+1 ṽ k+1 e T 2k, then there exists a trivial symplectic matrix D k =, ˆDk, ˆF k diagonal, such that V k = V k D k. [ ˆDk ˆFk ˆD 1 k ] Proof: Consequence of uniqueness of SR decomposition up to trivial symplectic factors and relation of SR decomposition to Krylov subspace. Peter Benner TU Chemnitz 18

19 choosing the free parameters Condition of Lanczos Basis Accuracy of computed eigenvalues is affected by condition of Lanczos basis. keep cond 2 (V n ) small (i.e., choose best trivial symplectic factor D k ) V n symplectic = V 1 n = JV T n J = cond 2 (V n ) = V n 2 2. V n 2 can be bounded by max { v k 2, w k 2 } V n 2 2n max { v k 2, w k 2 }, 1 k n 1 k n = Minimize τ := max 1 k n { v k 2, w k 2 }. Peter Benner TU Chemnitz 19

20 choosing the free parameters Optimizing One Free Parameter Fix ζ k such that v k 2 = 1 for all k = τ = max { w k 2 } = max { 1 1 k n 1 k n ν k Hv k δ k v k 2 }. Requiring symplectic Lanczos basis ν k is fixed. Choice of δ k independent of δ 1,..., δ k 1 = δ k = argmin{ Hv k δv k 2 }. δ R Peter Benner TU Chemnitz 20

21 choosing the free parameters Consider the quadratic form Minimizing τ f k (δ) = Hv k δv k 2 2 = (Hv k δv k ) T (Hv k δv k ) = vk T H T Hv k 2δvk T Hv k + δ 2 vk T v }{{ k. } =1 First-order necessary condition for a minimum is Hence, f k(δ) = 2v T k Hv k + 2δ = 0. δ k = v T k Hv k, i.e., v k w k (local orthogonality condition) Same choice as in [Ferng/Lin/Wang 97]. Peter Benner TU Chemnitz 21

22 numerical examples Numerical Examples I Control of string of high-speed vehicles: n = 398, v = e, different choices for δ j 3500 Condition of Lanczos basis δ = 1 δ = 0 orthogonality condition random δ number of Lanczos steps Peter Benner TU Chemnitz 22

23 numerical examples Numerical Examples II Random stable gyroscopic system: n = 200, different choices for δ j 15 Structured eigensolver QR algorithm Computed Eigenvalues 8 Condition of Lanczos basis Im(λ) δ = 1 δ = 0 orthogonality condition random δ Re(λ) x number of Lanczos steps Peter Benner TU Chemnitz 23

24 conclusions Conclusions Optimal choice of free parameters in symplectic Lanczos method can improve condition of Lanczos basis and may improve Ritz value accuracy. Optimization of two free parameters possible but... Thorough comparison necessary. For practical applications, use implicitly restarted variants [B./Faßbender 97, Watkins 02], but still more details necessary: shift-and-invert, deflation, locking, purging,... Variants for symplectic eigenproblem available [B./Faßbender 00]. Two-sided symplectic (implicitly restarted) Arnoldi based on symplectic URV decomposition under development [B./Kreßner/Mehrmann/Xu]. Peter Benner TU Chemnitz 24