Lagrangian Discretization of Variational Mean Field Games
Clément Sarrazin · SIAM Journal on Control and Optimization · 2022
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 envelope with 2-Wasserstein distance. Study of this envelope 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.