High order computation of optimal transport, mean field planning, and mean field games

TL;DR

This paper develops high-order numerical schemes using finite element methods to improve the computation of optimal transport, mean-field planning, and mean field games, demonstrating their efficiency and convergence through experiments.

Contribution

It introduces and validates high-order finite element methods for solving complex mean-field problems, a novel approach in this field.

Findings

High-order methods achieve faster convergence rates.

Numerical experiments confirm the efficiency of the proposed approach.

The methods effectively solve OT, MFP, and MFG problems.

Abstract

Mean-field games (MFGs) have shown strong modeling capabilities for large systems in various fields, driving growth in computational methods for mean-field game problems. However, high order methods have not been thoroughly investigated. In this work, we explore applying general high-order numerical schemes with finite element methods in the space-time domain for computing the optimal transport (OT), mean-field planning (MFP), and MFG problems. We conduct several experiments to validate the convergence rate of the high order method numerically. Those numerical experiments also demonstrate the efficiency and effectiveness of our approach.