On the convergence of Krylov methods with low-rank truncations

TL;DR

This paper investigates how low-rank truncations in Krylov methods influence convergence and provides guidelines to ensure convergence is maintained, supported by theoretical analysis and numerical experiments.

Contribution

It offers a theoretical framework for performing low-rank truncations in Krylov methods without losing convergence properties.

Findings

Proper truncation strategies preserve convergence.

Theoretical analysis confirms truncation impact on convergence.

Numerical experiments validate the proposed methods.

Abstract

Low-rank Krylov methods are one of the few options available in the literature to address the numerical solution of large-scale general linear matrix equations. These routines amount to well-known Krylov schemes that have been equipped with a couple of low-rank truncations to maintain a feasible storage demand in the overall solution procedure. However, such truncations may affect the convergence properties of the adopted Krylov method. In this paper we show how the truncation steps have to be performed in order to maintain the convergence of the Krylov routine. Several numerical experiments validate our theoretical findings.