Convergence Analysis for Nonlinear GMRES

TL;DR

This paper provides a convergence analysis for the nonlinear GMRES method, showing linear convergence of residuals under certain conditions for different window sizes, enhancing understanding of its theoretical properties.

Contribution

It offers the first convergence proofs for NGMRES with finite window size applied to nonlinear systems, filling a key gap in the theoretical understanding.

Findings

Residuals of NGMRES(m) converge r-linearly for m>0.

Residuals of NGMRES(0) converge q-linearly.

Theoretical convergence conditions are established.

Abstract

In this work, we revisit nonlinear generalized minimal residual method (NGMRES) applied to nonlinear problems. NGMRES is used to accelerate the convergence of fixed-point iterations, which can substantially improve the performance of the underlying fixed-point iterations. We consider NGMRES with a finite window size $m$, denoted as NGMRES($m$). However, there is no convergence analysis for NGMRES($m$) applied to nonlinear systems. We prove that for general $m>0$, the residuals of NGMRES($m$) converge r-linearly under some conditions. For $m=0$, we prove that the residuals of NGMRES(0) converge q-linearly.