Convergence analysis of Anderson acceleration for nonlinear equations with H\"older continuous derivatives

TL;DR

This paper analyzes the local convergence of Anderson acceleration for solving nonlinear equations with H"older continuous derivatives, providing theoretical convergence rates and demonstrating practical efficiency improvements in challenging large-scale problems.

Contribution

It establishes local R-linear convergence results for Anderson acceleration with general depth under H"older continuity, and demonstrates its effectiveness on complex nonlinear systems.

Findings

Anderson acceleration converges linearly under H"older continuous Jacobians.

Sharper convergence factors are obtained in the Lipschitz case.

Significant reductions in iterations and computation time are observed in large-scale applications.

Abstract

This work investigates the local convergence behavior of Anderson acceleration in solving nonlinear systems. We establish local R-linear convergence results for Anderson acceleration with general depth $m$ under the assumptions that the Jacobian of the nonlinear operator is H\"older continuous and the corresponding fixed-point function is contractive. In the Lipschitz continuous case, we obtain a sharper R-linear convergence factor. We also derive a refined residual bound for the depth $m = 1$ under the same assumptions used for the general depth results. Applications to a nonsymmetric Riccati equation from transport theory demonstrate that Anderson acceleration yields comparable results to several existing fixed-point methods for the regular cases, and that it brings significant reductions in both the number of iterations and computation time, even in challenging cases involving nearly singular or large-scale problems.