Newton Methods for Mean Field Games: A Numerical Study

TL;DR

This paper develops and tests Newton-based numerical methods for solving second-order Mean Field Game problems, demonstrating their robustness, accuracy, and efficiency through various benchmark experiments.

Contribution

It introduces new finite difference and semi-Lagrangian discretization schemes for Newton iterations in infinite dimensions, advancing computational methods for MFGs.

Findings

Newton methods exhibit quadratic convergence for MFGs

The proposed schemes are robust and accurate in benchmark tests

Comparative analysis shows advantages over existing methods

Abstract

We address the numerical solution of second-order Mean Field Game problems through Newton iterations in infinite dimensions, introduced in [14], where quadratic convergence of the method was rigorously established. Building upon this theoretical framework, we develop new numerical discretization techniques, including both a finite difference and a semi-Lagrangian scheme, that enable an effective computational implementation of the infinite-dimensional iterations. The proposed methods are tested on several benchmark problems, and the resulting numerical experiments demonstrate their robustness, accuracy, and efficiency. A comparative analysis between the two schemes and existing approaches from the literature is also presented, highlighting the potential of Newton-based solvers for MFG systems.