A semi-Lagrangian scheme for First-Order Mean Field Games based on monotone operators

TL;DR

This paper develops a semi-Lagrangian numerical scheme for first-order mean field games, utilizing monotone operators, and introduces a Learning Value Algorithm with acceleration strategies, validated through numerical experiments.

Contribution

It introduces a novel semi-Lagrangian scheme for mean field games and a convergent learning algorithm with acceleration techniques.

Findings

The scheme converges to a weak solution of the MFG system.

The Learning Value Algorithm is proven to converge.

Acceleration via Policy iteration significantly enhances computational performance.

Abstract

We construct a semi-Lagrangian scheme for first-order, time-dependent, and non-local Mean Field Games. The convergence of the scheme to a weak solution of the system is analyzed by exploiting a key monotonicity property. To solve the resulting discrete problem, we implement a Learning Value Algorithm, prove its convergence, and propose an acceleration strategy based on a Policy iteration method. Finally, we present numerical experiments that validate the effectiveness of the proposed schemes and show that the accelerated version significantly improves performance.