Numerical Methods for Large Scale Eigenvalue Problems

Transcription

1 MAX PLANCK INSTITUTE Summer School in Trogir, Croatia Oktober 12, 2011 Numerical Methods for Large Scale Eigenvalue Problems Patrick Kürschner Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 1/6

2 Eigenvalue Problems Introduction Eigenvalue problems Goal: find eigenvalues λ C and eigenvectors x C n \{0} solving standard eigenvalue problems Ax = λx, A C n n, nonlinear eigenvalue problems T (λ)x = 0, T : C C n n. Here, the defining matrices are large and sparse. Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 2/6

3 Eigenvalue Problems Application Relation to dynamical systems Consider the linear time invariant control system ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) The transfer function is H(s) = C(sI n A) 1 B. Its poles are the eigenvalues of T (λ) = λi n A (standard eigenvalue problem). Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 3/6

4 Eigenvalue Problems Application Relation to dynamical systems Consider the linear time invariant control system Mẍ(t) + Dẋ(t) + Kx(t) = Bu(t), y(t) = Cx(t) The transfer function is H(s) = C(s 2 M + sd + K) 1 B. Its poles are the eigenvalues of T (λ) = λ 2 M + λd + K (quadratic eigenvalue problem). Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 3/6

5 Eigenvalue Problems Application Relation to dynamical systems Consider the linear time invariant control system ẋ(t) = Ax(t) + Gx(t τ)bu(t), τ > 0 y(t) = Cx(t) The transfer function is H(s) = C(sI n A e τs G) 1 B. Its poles are the eigenvalues of T (λ) = λi n A e τλ G (delay eigenvalue problem). Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 3/6

6 Newton s Method Small Nonlinear Problems Note that an eigenpair (λ, x) of T (λ) is a root of the nonlinear function [ ] T (λ)x F (x, λ) = w H, F : C n+1 C n+1. x 1 First idea: apply Newton s method. Initial approximation (θ, v) (λ, x), Newton system for the next (hopefully better) approximation (θ +, v + ) is [ v+ θ + ] = [ v θ ] [ F (v, θ)] 1 F (v, θ). Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 4/6

7 Newton s Method Small Nonlinear Problems Note that an eigenpair (λ, x) of T (λ) is a root of the nonlinear function [ ] T (λ)x F (x, λ) = w H, F : C n+1 C n+1. x 1 First idea: apply Newton s method. Initial approximation (θ, v) (λ, x), Newton system for the next (hopefully better) approximation (θ +, v + ) is [ ] [ v+ v = θ + θ] [ ] 1 [ ] T (θ) Ṫ (θ)v T (λ)v w H 0 w H. v 1 Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 4/6

8 Newton s Method Small Nonlinear Problems Note that an eigenpair (λ, x) of T (λ) is a root of the nonlinear function [ ] T (λ)x F (x, λ) = w H, F : C n+1 C n+1. x 1 First idea: apply Newton s method. Initial approximation (θ, v) (λ, x), Newton system for the next (hopefully better) approximation (θ +, v + ) is [ ] [ v+ v = θ + θ] [ ] 1 [ ] T (θ) Ṫ (θ)v T (λ)v w H 0 w H. v 1 Drawbacks Requires good initial approximations (θ, v) Matrix inversion infeasible for large problems Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 4/6

9 Nonlinear Jacobi-Davidson Large-Scale Nonlinear Problems 1 Project the operator T (λ) onto a low-dimensional subspace V C n, dim(v) = k n V H T (λ) V = H(λ) Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 5/6

10 Nonlinear Jacobi-Davidson Large-Scale Nonlinear Problems 1 Project the operator T (λ) onto a low-dimensional subspace V C n, dim(v) = k n V H T (λ) V = H(λ) 2 Solve the small problem H(θ)q = 0, e.g., using Newton type methods or variations of thereof. Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 5/6

11 Nonlinear Jacobi-Davidson Large-Scale Nonlinear Problems 1 Project the operator T (λ) onto a low-dimensional subspace V C n, dim(v) = k n V H T (λ) V = H(λ) 2 Solve the small problem H(θ)q = 0, e.g., using Newton type methods or variations of thereof. 3 Expand V orthogonally by t v := Vq, obtained from the approximate solution of the Jacobi-Davidson correction equation ( I Ṫ (θ)vv H v HṪ (θ)v ) T (θ) ( I vv H) t = r = T (θ)v. Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 5/6

12 Nonlinear Jacobi-Davidson Large-Scale Nonlinear Problems 1 Project the operator T (λ) onto a low-dimensional subspace V C n, dim(v) = k n V H T (λ) V = H(λ) 2 Solve the small problem H(θ)q = 0, e.g., using Newton type methods or variations of thereof. 3 Expand V orthogonally by t v := Vq, obtained from the approximate solution of the Jacobi-Davidson correction equation ( I Ṫ (θ)vv H v HṪ (θ)v ) T (θ) ( I vv H) t = r = T (θ)v. 4 Repeat process with V = [V, t] until convergence. Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 5/6

13 Nonlinear Jacobi-Davidson Large-Scale Nonlinear Problems 1 Project the operator T (λ) onto a low-dimensional subspace V C n, dim(v) = k n V H T (λ) V = H(λ) 2 Solve the small problem H(θ)q = 0, e.g., using Newton type methods or variations of thereof. 3 Expand V orthogonally by t v := Vq, obtained from the approximate solution of the Jacobi-Davidson correction equation ( I Ṫ (θ)vv H v HṪ (θ)v ) T (θ) ( I vv H) t = r = T (θ)v. 4 Repeat process with V = [V, t] until convergence. Advantage Applies only cheap operations compared to Newton s method. Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 5/6

14 Nonlinear Jacobi-Davidson Large-Scale Nonlinear Problems Disadvantage Newton s type method: Still highly dependent on initial data. A lot of freedom w.r.t. program settings. Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 6/6

15 Nonlinear Jacobi-Davidson Large-Scale Nonlinear Problems Disadvantage Newton s type method: Still highly dependent on initial data. A lot of freedom w.r.t. program settings. How to solve the small nonlinear problem? How to solve the correction equation inexactly? what accuracy, which solution method (GMRES, QMR,...), what kind of preconditioner? Computation of several eigenpairs. Incorporation of left eigenvectors for faster convergence?... Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 6/6

16 Linear Jacobi-Davidson Large-Scale Linear Problems Linear Problems Ax = λbx: Solution strategies for some of the previous issues. Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 7/6

17 Linear Jacobi-Davidson Large-Scale Linear Problems Linear Problems Ax = λbx: Solution strategies for some of the previous issues. How to solve the small problem? Easy: QZ method. How to solve the correction equation inexactly? what accuracy, solution method: criteria for controlling the inner accuracy w.r.t. outer accucary. what kind of preconditioner? E.g. ilu of A τb. Computation of several eigenpairs: Orthogonalize against found eigenvectors. Incorporation of left eigenvectors for faster convergence? Two-sided Jacobi-Davidson... Max Planck Institute Magdeburg Patrick Kürschner, Numerical Methods for Large Scale Eigenvalue Problems 7/6