Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization

TL;DR

This paper extends the preconditioned inverse iteration method to operators for computing the smallest eigenvalue of elliptic problems, demonstrating convergence with approximate applications and showcasing adaptive wavelet discretization benefits.

Contribution

It introduces the perturbed preconditioned inverse iteration (PPINVIT) for operators, providing convergence analysis and demonstrating advantages of adaptive wavelet methods over uniform refinement.

Findings

PPINVIT converges up to any tolerance with approximate operator applications.

Adaptive wavelet discretization outperforms uniform refinement for eigenfunction approximation.

Numerical experiments confirm theoretical convergence and efficiency.

Abstract

In this paper we discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and geometric proof based on the preconditioned inverse iteration (PINVIT) for matrices [Knyazev and Neymeyr, (2009)] is extended to the case of operators. We show that convergence is retained up to any tolerance if one only uses approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). We then analyze the Besov regularity of the eigenfunctions of the Poisson eigenvalue problem on a polygonal domain, showing the advantage of an adaptive solver to uniform refinement when using a stable wavelet base. A numerical example for PPINVIT, applied to the model problem on the L-shaped domain, is shown to reproduce the predicted behaviour.