Convergence Properties of Nonlinear GMRES Applied to Linear Systems

TL;DR

This paper analyzes the convergence of Nonlinear GMRES when applied to linear systems, revealing its relationship with classical GMRES and providing theoretical insights into its effectiveness.

Contribution

It offers the first convergence analysis of NGMRES for linear systems and establishes equivalences with classical GMRES, advancing understanding of its theoretical properties.

Findings

NGMRES converges under certain conditions similar to GMRES

Identifies equivalences between NGMRES and classical GMRES

Provides theoretical bounds for NGMRES convergence

Abstract

The Nonlinear GMRES (NGMRES) proposed by Washio and Oosterlee [Electron. Trans. Numer. Anal, 6(271-290), 1997] is an acceleration method for fixed point iterations. It has been demonstrated to be effective, but its convergence properties have not been extensively studied in the literature so far. In this work we aim to close some of this gap, by offering a convergence analysis for NGMRES applied to linear systems. A central part of our analysis focuses on identifying equivalences between NGMRES and the classical Krylov subspace GMRES method.