Convergence Analysis of the Alternating Anderson-Picard Method for Nonlinear Fixed-point Problems

TL;DR

This paper analyzes the convergence of the Alternating Anderson-Picard (AAP) method for nonlinear fixed-point problems, establishing its relation to multisecant-GMRES and Newton-GMRES methods, and proving local q-linear convergence.

Contribution

The paper provides the first theoretical analysis of AAP in the nonlinear case, showing its equivalence to multisecant-GMRES and convergence to Newton-GMRES.

Findings

AAP converges to Newton-GMRES as residual approaches zero

AAP is locally q-linear convergent with an explicit convergence bound

Numerical examples validate the theoretical convergence results

Abstract

Anderson Acceleration (AA) has been widely used to solve nonlinear fixed-point problems due to its rapid convergence. This work focuses on a variant of AA in which multiple Picard iterations are performed between each AA step, referred to as the Alternating Anderson-Picard (AAP) method. Despite introducing more ``slow'' Picard iterations, this method has been shown to be efficient and even more robust in both linear and nonlinear cases. However, there is a lack of theoretical analysis for AAP in the nonlinear case. In this paper, we address this gap by establishing the equivalence between AAP and a multisecant-GMRES method that uses GMRES to solve a multisecant linear system at each iteration. From this perspective, we show that AAP ``converges'' to the Newton-GMRES method. Specifically, as the residual approaches zero, the multisecant matrix, the approximate Jacobian inverse, the search direction, and the optimization gain of AAP converge to their counterparts in the Newton-GMRES method. These connections provide insights for analyzing the asymptotic convergence properties of AAP. Consequently, we show that AAP is locally $q$-linear convergent and provide an upper bound for the convergence factor of AAP. To validate the theoretical results, numerical examples are provided.