One-step convergence of inexact Anderson acceleration for contractive and non-contractive mappings

Abstract

We give a one-step convergence analysis of inexact Anderson acceleration for the fixed point iteration x k+1 = g(x k ) with a potentially non-contractive mapping g, where g(x k ) is evaluated approximately and the minimization of the nonlinear residual norms is performed in the vector 2-norm by the linear least-squares method. If g is non-contractive, then the original fixed point iteration does not converge, but a recent analysis by S. Pollock and L. Rebholz [IMA J. Numer. Anal., 41 (2021), pp. 2841-2872 shows that Anderson acceleration may still converge provided that the minimization at each step has a sufficient gain. In this paper, we show that inexact Anderson acceleration exhibits essentially the same convergence behavior as the exact algorithm if each g(x k ) is evaluated with an error proportional to the nonlinear residual norm w k = g(x k ) − x k , regardless of whether g is contractive or not. This means that the existing relationship between exact and inexact Anderson acceleration can be generalized in a unified framework for both contractive and non-contractive mappings. Numerical experiments show that the inexact algorithm can converge as rapidly as the exact counterpart while it can lower the computational cost.