Some theoretical results on the finite convergence property and the temporary stalling behavior of Anderson acceleration on linear systems

Abstract

We consider Anderson acceleration with window size $m$ (AA($m$)) applied to fixed-point iteration for linear systems. We explore some conditions on the $m+1$ initial guesses of AA($m$), aiming for the residuals $r_{m+1}=0$. We propose the sufficient and necessary conditions on the $m+1$ initial guesses for $r_{m+1}=0$. These findings can help us better understand the performance between the original fixed-point iteration and Anderson acceleration. Moreover, under some conditions, sharp lower and upper bounds on the coefficients of AA(1) for both linear and nonlinear systems have been proposed. A recent work [H. De Sterck et al., J. Sci. Comput. 99(1), 12, 2024] demonstrates an instance where the temporary stalling behavior $r_2=r_1$ occurs for AA(1). However, it leaves the question of the existence of $r_{k_0}=r_{k_0-1}\ne 0$ for some integer $k_0>2$ of AA(1). We give examples to show the temporary stalling behavior of Anderson acceleration for $k_0>2$ applied to solving linear systems.