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Numerical Methods for LargeScale Eigenvalue Problems Patrick K - - PowerPoint PPT Presentation

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Summer School in Trogir, Croatia Oktober 12, 2011 Numerical Methods for LargeScale Eigenvalue Problems Patrick K urschner Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory

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MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG

Summer School in Trogir, Croatia Oktober 12, 2011

Numerical Methods for Large–Scale Eigenvalue Problems

Patrick K¨ urschner

Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 1/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Eigenvalue Problems

Introduction

Eigenvalue problems

Goal: find eigenvalues λ ∈ C and eigenvectors x ∈ Cn\{0} solving standard eigenvalue problems Ax = λx, A ∈ Cn×n, nonlinear eigenvalue problems T(λ)x = 0, T : C → Cn×n. Here, the defining matrices are large and sparse.

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 2/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Eigenvalue Problems

Application

Relation to dynamical systems

Consider the linear time invariant control system ˙ x(t) = Ax(t) + Bu(t), y(t) = Cx(t) The transfer function is H(s) = C(sIn − A)−1B. Its poles are the eigenvalues of T(λ) = λIn − A (standard eigenvalue problem).

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 3/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Eigenvalue Problems

Application

Relation to dynamical systems

Consider the linear time invariant control system M¨ x(t) + D ˙ x(t) + Kx(t) = Bu(t), y(t) = Cx(t) The transfer function is H(s) = C(s2M + sD + K)−1B. Its poles are the eigenvalues of T(λ) = λ2M + λD + K (quadratic eigenvalue problem).

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 3/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Eigenvalue Problems

Application

Relation to dynamical systems

Consider the linear time invariant control system ˙ x(t) = Ax(t) + Gx(t − τ)Bu(t), τ > 0 y(t) = Cx(t) The transfer function is H(s) = C(sIn − A − e−τsG)−1B. Its poles are the eigenvalues of T(λ) = λIn − A − e−τλG (delay eigenvalue problem).

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 3/6

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SLIDE 6

Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Newton’s Method

Small Nonlinear Problems

Note that an eigenpair (λ, x) of T(λ) is a root of the nonlinear function F(x, λ) = T(λ)x w Hx − 1

  • , F : Cn+1 → Cn+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 K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 4/6

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SLIDE 7

Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Newton’s Method

Small Nonlinear Problems

Note that an eigenpair (λ, x) of T(λ) is a root of the nonlinear function F(x, λ) = T(λ)x w Hx − 1

  • , F : Cn+1 → Cn+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

  • −
  • T(θ)

˙ T(θ)v w H −1 T(λ)v w Hv − 1

  • .

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 4/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Newton’s Method

Small Nonlinear Problems

Note that an eigenpair (λ, x) of T(λ) is a root of the nonlinear function F(x, λ) = T(λ)x w Hx − 1

  • , F : Cn+1 → Cn+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

  • −
  • T(θ)

˙ T(θ)v w H −1 T(λ)v w Hv − 1

  • .
  • Requires good initial approximations (θ, v)
  • Matrix inversion infeasible for large problems

Drawbacks

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 4/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Nonlinear Jacobi-Davidson

Large-Scale Nonlinear Problems

1

Project the operator T(λ) onto a low-dimensional subspace V ⊂ Cn, dim(V) = k ≪ n V H T(λ) V = H(

λ )

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 5/6

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SLIDE 10

Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Nonlinear Jacobi-Davidson

Large-Scale Nonlinear Problems

1

Project the operator T(λ) onto a low-dimensional subspace V ⊂ Cn, 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 K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 5/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Nonlinear Jacobi-Davidson

Large-Scale Nonlinear Problems

1

Project the operator T(λ) onto a low-dimensional subspace V ⊂ Cn, 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 −

˙ T(θ)vv H v H ˙ T(θ)v

  • T(θ)
  • I − vv H

t = −r = −T(θ)v.

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 5/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Nonlinear Jacobi-Davidson

Large-Scale Nonlinear Problems

1

Project the operator T(λ) onto a low-dimensional subspace V ⊂ Cn, 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 −

˙ T(θ)vv H v H ˙ T(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 K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 5/6

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SLIDE 13

Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Nonlinear Jacobi-Davidson

Large-Scale Nonlinear Problems

1

Project the operator T(λ) onto a low-dimensional subspace V ⊂ Cn, 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 −

˙ T(θ)vv H v H ˙ T(θ)v

  • T(θ)
  • I − vv H

t = −r = −T(θ)v.

4

Repeat process with V = [V , t] until convergence. Applies only cheap operations compared to Newton’s method. Advantage

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 5/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Nonlinear Jacobi-Davidson

Large-Scale Nonlinear Problems

Newton’s type method:

  • Still highly dependent on initial data.
  • A lot of freedom w.r.t. program settings.

Disadvantage

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 6/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Nonlinear Jacobi-Davidson

Large-Scale Nonlinear Problems

Newton’s type method:

  • Still highly dependent on initial data.
  • A lot of freedom w.r.t. program settings.

Disadvantage 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 K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 6/6

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Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Linear Jacobi-Davidson

Large-Scale Linear Problems

Ax = λBx:

  • Solution strategies for some of the previous issues.

Linear Problems

Max Planck Institute Magdeburg Patrick K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 7/6

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SLIDE 17

Eigenvalue Problems Newton’s Method Nonlinear Jacobi-Davidson

Linear Jacobi-Davidson

Large-Scale Linear Problems

Ax = λBx:

  • Solution strategies for some of the previous issues.

Linear Problems 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 K¨ urschner, Numerical Methods for Large–Scale Eigenvalue Problems 7/6