Spectral discretization errors in filtered subspace iteration

TL;DR

This paper analyzes how spectral discretization errors affect the accuracy of filtered subspace iteration in approximating eigenvalues and eigenspaces of selfadjoint operators, providing bounds and numerical validation.

Contribution

It introduces an abstract framework for quantifying discretization errors in filtered subspace iteration and demonstrates its application with finite element discretization for elliptic operators.

Findings

Bounds for Hausdorff distance between true and computed eigenvalue clusters

Discretization errors can be effectively estimated within the proposed framework

Numerical experiments confirm the sharpness of theoretical bounds

Abstract

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates.