Filtering for Anderson acceleration

TL;DR

This paper presents a filtering strategy for Anderson acceleration that improves the condition of the least-squares problem at each iteration, enhancing convergence especially for challenging initial conditions.

Contribution

It introduces a novel two-step filtering method that controls matrix column disparity and subspace angles, ensuring better numerical stability in Anderson acceleration.

Findings

Effective on PDE discretization problems

Improves convergence for difficult initial guesses

Controls condition number of least-squares matrices

Abstract

This work introduces, analyzes and demonstrates an efficient and theoretically sound filtering strategy to ensure the condition of the least-squares problem solved at each iteration of Anderson acceleration. The filtering strategy consists of two steps: the first controls the length disparity between columns of the least-squares matrix, and the second enforces a lower bound on the angles between subspaces spanned by the columns of that matrix. The combined strategy is shown to control the condition number of the least-squares matrix at each iteration. The method is shown to be effective on a range of problems based on discretizations of partial differential equations. It is shown particularly effective for problems where the initial iterate may lie far from the solution, and which progress through distinct preasymptotic and asymptotic phases.