Lagrangian discretization of variational mean field games

In this article, we introduce a method to approximate solutions of some variational mean field game problems with congestion, by finite sets of player trajectories. These trajectories are obtained by solving a minimization problem similar to the initial variational problem. In this discretized problem, congestion is penalized by a Moreau envelop with the 2-Wasserstein distance. Study of this envelop as well as efficient computation of its values and variations is done using semi-discrete optimal transport. We show convergence of the discrete sets of trajectories toward a solution of the mean field game, under some conditions on the parameters of the discretization.