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.

本文研究具有本地耦合的时变均值场博弈系统的数值逼近，使用基于变分方法的离散化方法，并应用 Chambolle 和 Pock 介绍的原始 - 对偶算法来解决其有限维变分问题，特别是使用适当的预处理迭代算法来改善解决线性系统的方法。