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

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.