Stability of linear GMRES convergence with respect to compact perturbations

TL;DR

This paper analyzes how the convergence of GMRES for a linear operator is affected by compact perturbations, extending previous results to more general operators and providing bounds based on singular values.

Contribution

It extends known GMRES convergence bounds to operators with compact perturbations, generalizing earlier results limited to scalar multiples of the identity.

Findings

GMRES convergence bound depends on singular values of the perturbation

Results extend superlinear convergence analysis to broader classes of operators

Provides explicit bounds for convergence degradation under compact perturbations

Abstract

Suppose that a linear bounded operator $B$ on a Hilbert space exhibits at least linear GMRES convergence, i.e., there exists $M_B<1$ such that the GMRES residuals fulfill $\|r_k\|\leq M_B\|r_{k-1}\|$ for every initial residual $r_0$ and step $k\in\mathbb{N}$. We prove that GMRES with a compactly perturbed operator $A=B+C$ admits the bound $\|r_k\|/\|r_0\|\leq\prod_{j=1}^k\bigl(M_B+(1+M_B)\,\|A^{-1}\|\,\sigma_j(C)\bigr)$, i.e., the singular values $\sigma_j(C)$ control the departure from the bound for the unperturbed problem. This result can be seen as an extension of [I. Moret, A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal., 34 (1997), pp. 513-516, https://doi.org/10.1137/S0036142993259792], where only the case $B=\lambda I$ is considered. In this special case $M_B=0$ and the resulting convergence is superlinear.