Abstract
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, as well as conditions on the discretization in order to get this convergence. We also show strong convergence (in particular almost everywhere) in some cases for the underlying densities resulting from the Moreau projection of solution to the discrete problem.
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