On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings

TL;DR

This paper develops a primal-dual algorithm for numerically solving time-dependent mean field games with local couplings, improving efficiency for large viscosity parameters through preconditioned iterative solvers.

Contribution

It extends a primal-dual algorithm to time-dependent MFGs and introduces preconditioned iterative methods for large viscosity parameters, enhancing computational efficiency.

Findings

Effective implementation of primal-dual algorithm for time-dependent MFGs.

Improved linear system solutions with preconditioning for large viscosity.

Enhanced numerical approximation accuracy and efficiency.

Abstract

We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [SIAM J. Numer. Anal., 48(3):1136-1162, 2010]. In order to solve the finite dimensional variational problems, in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [J. Math. Imaging Vision, 40(1):120-145, 2011], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] by suitable preconditioned iterative algorithms.