A short report on preconditioned Anderson acceleration method

TL;DR

This paper introduces a flexible preconditioned Anderson acceleration (PAA) method that enhances fixed-point iteration convergence by combining acceleration techniques with preconditioning, adaptable to various nonlinear problems.

Contribution

The paper presents a novel PAA framework that unifies and extends existing fixed-point methods, offering adjustable convergence and computational efficiency.

Findings

PAA improves convergence rates across different problems.

Preconditioning strategies significantly enhance robustness.

Delayed updates reduce computational costs effectively.

Abstract

In this report, we present a versatile and efficient preconditioned Anderson acceleration (PAA) method for fixed-point iterations. The proposed framework offers flexibility in balancing convergence rates (linear, super-linear, or quadratic) and computational costs related to the Jacobian matrix. Our approach recovers various fixed-point iteration techniques, including Picard, Newton, and quasi-Newton iterations. The PAA method can be interpreted as employing Anderson acceleration (AA) as its own preconditioner or as an accelerator for quasi-Newton methods when their convergence is insufficient. Adaptable to a wide range of problems with differing degrees of nonlinearity and complexity, the method achieves improved convergence rates and robustness by incorporating suitable preconditioners. We test multiple preconditioning strategies on various problems and investigate a delayed update strategy for preconditioners to further reduce the computational costs.