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FEAST for differential eigenvalue problems

Andrew Horning, Alex Townsend

TL;DR

An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane.

Abstract

An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane. Unbounded search regions are handled with a novel rational filter for the right half-plane. If the differential operator is normal or self-adjoint, then the operator analogue preserves that structure and robustly computes eigenvalues to near machine precision accuracy. The algorithm is particularly adept at computing high-frequency modes of differential operators that possess self-adjoint structure with respect to weighted Hilbert spaces.

FEAST for differential eigenvalue problems

TL;DR

An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane.

Abstract

An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane. Unbounded search regions are handled with a novel rational filter for the right half-plane. If the differential operator is normal or self-adjoint, then the operator analogue preserves that structure and robustly computes eigenvalues to near machine precision accuracy. The algorithm is particularly adept at computing high-frequency modes of differential operators that possess self-adjoint structure with respect to weighted Hilbert spaces.

Paper Structure

This paper contains 18 sections, 5 theorems, 54 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. The FEAST matrix eigensolver
  3. An operator analogue of FEAST
  4. FEAST for differential operators
  5. Condition number of the Ritz values
  6. Pseudospectra of $\mathbf{Q^*\mathcal{L}Q}$
  7. A practical differential eigensolver based on an operator analogue of FEAST
  8. Computing high-frequency eigenmodes
  9. A regular Sturm--Liouville eigenvalue problem
  10. An indefinite Sturm--Liouville eigenvalue problem
  11. Convergence and stability
  12. Rational subspace iteration for differential operators
  13. A pseudospectral inclusion theorem
  14. An operator analogue of the Rayleigh Quotient Iteration
  15. Free vibrations of an airplane wing
  16. ...and 3 more sections

Key Result

Theorem 1

\newlabelthm:condition number0 Let $\mathcal{L}:\mathcal{D}(\mathcal{L})\rightarrow\mathcal{H}$ be a closed and densely defined operator on a Hilbert space $\mathcal{H}$, $Q:\mathbb{C}^m\rightarrow\mathcal{H}$ be an invariant subspace of $\mathcal{L}$ satisfying $Q^*Q=I$, and $L=Q^*\mathcal{L}Q$.

Figures (7)

  • Figure 1: Left: The eigenvalue condition numbers ANGUAS2019170 for $4000\times 4000$ discretizations of \ref{['eqn:oscillator']} obtained by collocation (blue dots), tau (red dots), Chebyshev--Galerkin (black dots), and ultraspherical (yellow dots) spectral methods are compared to the eigenvalue condition numbers (magenta dots) of \ref{['eqn:oscillator']}, which are preserved by the operator analogue of FEAST. Right: The relative errors in the first $2000$ eigenvalues of each spectral discretization of \ref{['eqn:oscillator']}, computed with a backward stable eigensolver golub2012matrix. We observe fluctuations in the relative errors due to the ill-conditioning introduced by using nonsymmetric spectral discretizations of $\mathcal{L}$. In contrast, the relative errors (magenta dots) in the eigenvalues computed by contFEAST, a practical implementation of the operator analogue of FEAST (see \ref{['sec:practical algorithm']}), are on the order of machine precision.
  • Figure 1: Left: FEAST uses an approximation to the spectral projector to compute the eigenvalues that lie inside $\Omega$ (red dots) and project away the eigenvalues outside of $\Omega$ (blue dots). Right: The rational map in \ref{['eqn:approx_FEAST_proj']} that approximates the characteristic function on $\Omega$.
  • Figure 1: Left: The large eigenvalues of \ref{['eqn:regular SLEP']} are computed by contFEAST (see Algorithm \ref{['alg:contFEAST']}) using search regions given by asymptotic estimates for the eigenvalues \ref{['eqn:SLEP asymptotics']}. Right: The relative difference $\lvert\hat{\lambda}_n-\lambda^{asy}_n\rvert/\lambda^{asy}_n$ between the eigenvalues $\hat{\lambda}_n$ computed by contFEAST and the asymptotic values $\lambda^{asy}_n$ from \ref{['eqn:SLEP asymptotics']}. The difference is compared to an $\mathcal{O}(n^{-2})$ relative error estimate akbarfam2004higher.
  • Figure 1: Selected free-vibration modes of an airplane wing modeled by \ref{['eqn:cantilever beam']}.
  • Figure 1: Left: The region $\Omega_R$ from \ref{['eqn:half-disk']} used in the derivation of the rational filter over the right half-plane. Right: The constructed rational filter \ref{['eqn:HP RF']} for the right half-plane with $\ell=20$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Proof 1
  • Theorem 2
  • Proof 2
  • Theorem 1
  • Theorem 2
  • Proof 3
  • Lemma 3
  • Proof 4