Benchmarking results for the Newton-Anderson method

TL;DR

This paper presents numerical benchmarking of the Anderson accelerated Newton method, demonstrating superlinear convergence on various problems and analyzing its convergence domain, including for degenerate cases.

Contribution

It provides the first comprehensive benchmarking results for the Newton-Anderson method, including convergence analysis for degenerate and nondegenerate problems.

Findings

Superlinear convergence demonstrated on benchmark problems

Convergence domain similar to Newton's method for degenerate cases

Potential for different solutions under slight perturbations

Abstract

This paper primarily presents numerical results for the Anderson accelerated Newton method on a set of benchmark problems. The results demonstrate superlinear convergence to solutions of both degenerate and nondegenerate problems. The convergence for nondegenerate problems is also justified theoretically. For degenerate problems, those whose Jacobians are singular at a solution, the domain of convergence is studied. It is observed in that setting that Newton-Anderson has a domain of convergence similar to Newton, but it may be attracted to a different solution than Newton if the problems are slightly perturbed.