Newton-Anderson at Singular Points

TL;DR

This paper analyzes the convergence of Anderson acceleration applied to Newton's method near singular Jacobians, introducing a safeguarding strategy and demonstrating improved convergence on benchmark problems.

Contribution

It provides a theoretical convergence analysis for Newton-Anderson at singular points and proposes a new safeguarding strategy to enhance stability.

Findings

Anderson acceleration improves convergence near singular Jacobians.

The proposed safeguarding strategy enhances robustness of the method.

Numerical experiments confirm theoretical results on benchmark problems.

Abstract

In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton's method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and some standard benchmark examples.