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A Krylov Eigenvalue Solver Based on Filtered Time Domain Solutions

Lothar Nannen, Markus Wess

TL;DR

The paper addresses the challenge of computing eigenpairs for large-scale generalized Hermitian problems $S v = \omega^2 M v$ by integrating explicit time-domain simulation of the associated wave equation with a Fourier-inspired filter into a Krylov framework. By constructing a filter operator $C=\tilde{β}_α(M^{-1}S)$ from time-stepping the wave equation and applying it to build a Krylov basis, the method projects the large problem onto a small subspace and solves a manageable projected eigenproblem, avoiding costly inverses. Numerical experiments in 2D and on a large 3D horn-in-a-room model validate convergence and illustrate parameter effects, showing the approach can reliably target resonances within a specified region of interest despite dense spectra. The approach is flexible: it accommodates various time-stepping schemes, weight-function designs, and solver variants, and is particularly suited for very large problems where direct solvers or shift-invert strategies are impractical. Overall, the method provides a scalable, preprocessing-light pathway to compute interior or clustered eigenvalues in large-scale finite-element discretizations of wave-type operators, with clear potential for extension to open systems and different elliptic operators.

Abstract

This paper introduces a method for computing eigenvalues and eigenvectors of a generalized Hermitian, matrix eigenvalue problem. The work is focused on large scale eigenvalue problems, where the application of a direct inverse is out of reach. Instead, an explicit time-domain integrator for the corresponding wave problem is combined with a proper filtering and a Krylov iteration in order to solve for eigenvalues within a given region of interest. We report results of small scale model problems to confirm the reliability of the method, as well as the computation of acoustic resonances in a three dimensional model of a hunting horn to demonstrate the efficiency.

A Krylov Eigenvalue Solver Based on Filtered Time Domain Solutions

TL;DR

The paper addresses the challenge of computing eigenpairs for large-scale generalized Hermitian problems by integrating explicit time-domain simulation of the associated wave equation with a Fourier-inspired filter into a Krylov framework. By constructing a filter operator from time-stepping the wave equation and applying it to build a Krylov basis, the method projects the large problem onto a small subspace and solves a manageable projected eigenproblem, avoiding costly inverses. Numerical experiments in 2D and on a large 3D horn-in-a-room model validate convergence and illustrate parameter effects, showing the approach can reliably target resonances within a specified region of interest despite dense spectra. The approach is flexible: it accommodates various time-stepping schemes, weight-function designs, and solver variants, and is particularly suited for very large problems where direct solvers or shift-invert strategies are impractical. Overall, the method provides a scalable, preprocessing-light pathway to compute interior or clustered eigenvalues in large-scale finite-element discretizations of wave-type operators, with clear potential for extension to open systems and different elliptic operators.

Abstract

This paper introduces a method for computing eigenvalues and eigenvectors of a generalized Hermitian, matrix eigenvalue problem. The work is focused on large scale eigenvalue problems, where the application of a direct inverse is out of reach. Instead, an explicit time-domain integrator for the corresponding wave problem is combined with a proper filtering and a Krylov iteration in order to solve for eigenvalues within a given region of interest. We report results of small scale model problems to confirm the reliability of the method, as well as the computation of acoustic resonances in a three dimensional model of a hunting horn to demonstrate the efficiency.
Paper Structure (19 sections, 2 theorems, 32 equations, 12 figures, 1 algorithm)

This paper contains 19 sections, 2 theorems, 32 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Presentation of the method
  3. Krylov eigenvalue solver
  4. Filtered time-domain solutions
  5. Discretization of the filtered time-domain solution
  6. Algorithm
  7. Numerical experiments
  8. Small scale (2d) examples
  9. Description of the experiments
  10. First results
  11. Different filters
  12. Large scale examples
  13. Description of the problem
  14. Generalizations and extensions
  15. Discrete Problem
  16. ...and 4 more sections

Key Result

Lemma 2.2

Let $(\omega^2,v)$ be an eigenpair of def:basisMatrixEVP and the filter function $\beta_\alpha:[0,\infty) \to \mathbb{R}$ be defined by Then $(\beta_\alpha(\omega),v)$ is an eigenpair of $\Pi_{\alpha}$, i.e., $\Pi_{\alpha} v=\beta_\alpha(\omega) v$. Vice versa, if $(\lambda,v)$ is an eigenpair of $\Pi_{\alpha}$, then there exists at least one eigenvalue $\omega^2$ of def:basisMatrixEVP such that

Figures (12)

  • Figure 1: Discrete filter functions for fixed time-step $\tau=0.025$, the target interval $[\omega_{\mathrm{min}},\omega_{\mathrm{max}}]=[2,4]$, and varying end times $T$.
  • Figure 2: Sketch of the two-dimensional geometry used for the small scale examples from Section \ref{['sec:small_scale_numerics']}
  • Figure 3: Exact resonances of \ref{['def:basisMatrixEVP']} on the circles with radii $r_l=1.5,r_r=0.15$ and the whole dumbbell domain (cf. Figure \ref{['fig:dumbbell_geos']} and \ref{['eq:dumbbell_params']}).
  • Figure 4: Numerical results for the computation of resonances of \ref{['def:basisMatrixEVP']} for the dumbbell domain and discrete filter function for $T=300\tau$, $\omega_{\mathrm{min}} = 0,\omega_{\mathrm{max}}=3$.
  • Figure 5: Errors and residuals for selected resonances of Fig. \ref{['fig:dumbbell_td_evs_qual']}.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 3.1