How Descriptive are GMRES Convergence Bounds?

TL;DR

This paper critically examines the effectiveness of various GMRES convergence bounds based on polynomial approximation in the spectrum, field of values, and pseudospectra, highlighting their strengths and limitations through examples.

Contribution

It compares and analyzes different GMRES convergence bounds, proposing adaptations for matrices with specific structures and discussing how pseudospectra approximations can inform convergence estimates.

Findings

Bounds vary in success depending on matrix properties

Pseudospectra-based bounds can be estimated during iteration

Adaptations improve bounds for matrices with low-dimensional invariant subspaces

Abstract

GMRES is a popular Krylov subspace method for solving linear systems of equations involving a general non-Hermitian coefficient matrix. The conventional bounds on GMRES convergence involve polynomial approximation problems in the complex plane. Three popular approaches pose this approximation problem on the spectrum, the field of values, or pseudospectra of the coefficient matrix. We analyze and compare these bounds, illustrating with six examples the success and failure of each. When the matrix departs from normality due only to a low-dimensional invariant subspace, we discuss how these bounds can be adapted to exploit this structure. Since the Arnoldi process that underpins GMRES provides approximations to the pseudospectra, one can estimate the GMRES convergence bounds as an iteration proceeds.