A Simple Multiscale Method for Mean Field Games

TL;DR

This paper introduces a multiscale numerical method for solving mean field games that improves convergence speed and applicability to local and nonlocal problems across different dimensions.

Contribution

It presents a novel multiscale approach that constructs solutions from coarse to fine grids, incorporating relaxation and higher-order schemes for enhanced accuracy.

Findings

Accelerates convergence of mean field game solutions

Applicable to both local and nonlocal problems

Demonstrates efficiency in 1D and 2D numerical examples

Abstract

This paper proposes a multiscale method for solving the numerical solution of mean field games which accelerates the convergence and addresses the problem of determining the initial guess. Starting from an approximate solution at the coarsest level, the method constructs approximations on successively finer grids via alternating sweeping, which not only allows for the use of classical time marching numerical schemes but also enables applications to both local and nonlocal problems. At each level, numerical relaxation is used to stabilize the iterative process. A second-order discretization scheme is derived for higher-order convergence. Numerical examples are provided to demonstrate the efficiency of the proposed method in both local and nonlocal, 1-dimensional and 2-dimensional cases.