On the Convergence of CROP-Anderson Acceleration Method

TL;DR

This paper investigates the convergence properties and relationships between Anderson Acceleration and the CROP algorithm, providing insights that could improve the efficiency of fixed-point iteration methods in scientific computations.

Contribution

It offers a detailed analysis of the convergence behavior of Anderson Acceleration and CROP, clarifying their connections and potential for enhanced computational performance.

Findings

CROP outperforms classical Anderson Acceleration in efficiency.

The convergence properties of both methods are closely related.

Numerical examples validate the theoretical insights.

Abstract

Anderson Acceleration is a well-established method that allows to speed up or encourage convergence of fixed-point iterations. It has been successfully used in a variety of applications, in particular within the Self-Consistent Field (SCF) iteration method for quantum chemistry and physics computations. In recent years, the Conjugate Residual with OPtimal trial vectors (CROP) algorithm was introduced and shown to have a better performance than the classical Anderson Acceleration with less storage needed. This paper aims to delve into the intricate connections between the classical Anderson Acceleration method and the CROP algorithm. Our objectives include a comprehensive study of their convergence properties, explaining the underlying relationships, and substantiating our findings through some numerical examples. Through this exploration, we contribute valuable insights that can enhance the understanding and application of acceleration methods in practical computations, as well as the developments of new and more efficient acceleration schemes.