Shanks and Anderson-type acceleration techniques for systems of nonlinear equations

TL;DR

This paper reviews and extends acceleration techniques for solving nonlinear systems, especially focusing on Shanks and Anderson methods, establishing convergence results and testing on PDE-related problems.

Contribution

It introduces modifications to Shanks transformations, provides a unifying framework for acceleration methods, and analyzes convergence of a stabilized Anderson method.

Findings

Modified Shanks transformations improve convergence.

A new convergence proof for stabilized Anderson acceleration.

Successful application to nonlinear PDE problems.

Abstract

This paper examines a number of extrapolation and acceleration methods, and introduces a few modifications of the standard Shanks transformation that deal with general sequences. One of the goals of the paper is to lay out a general framework that encompasses most of the known acceleration strategies. The paper also considers the Anderson Acceleration method under a new light and exploits a connection with quasi-Newton methods, in order to establish local linear convergence results of a stabilized version of Anderson Acceleration method. The methods are tested on a number of problems, including a few that arise from nonlinear Partial Differential Equations.