Convergence analysis of the Alternating Anderson-Picard method for nonlinear fixed-point problems (Xue Feng, UCLA)

Abstract: Anderson Acceleration (AA) has been widely used to solve nonlinear fixed-point problems due to its rapid convergence. This talk focuses on a variant of AA in which multiple Picard iterations are performed between each AA step, referred to as the Alternating Anderson-Picard (AAP) method. Despite introducing more `slow’ Picard iterations, this method has been demonstrated to be efficient and even more robust in both linear and nonlinear cases. However, there is a lack of theoretical analysis for AAP in the nonlinear context. In this work, we address this gap by establishing the equivalence between AAP and a multisecant-GMRES method that employs GMRES to solve a multisecant linear system at each iteration. From this perspective, we show that AAP actually “converges” the well-known Newton-GMRES method. These connections also help us understand the convergence behavior of AAP, especially the asymptotic convergence rate.