Section: New Results

Mean Field Games and augmented lagrangian methods for optimal transport

Roman Andreev

We apply the augmented Lagrangian method to the convex optimization problem of the instationary variational mean field games with diffusion. The system is first discretized with space-time tensor product piecewise polynomial bases. This leads to a sequence of linear problems posed on the space-time cylinder that are second order in the temporal variable and fourth order in the spatial variable. To solve these large linear problems with the preconditioned conjugate gradients method we propose a parameter-robust preconditioner that is based on a temporal transformation coupled with a spatial multigrid. Numerical examples illustrate the method. [27].

G. Carlier J-D. Benamou have written in collaboration with F. Santambrogio a review paper on variational MFG [31] both on theoretical and numerical aspects, the latter being addressed by augmented Lagrangian techniques developed by our team also in the context of optimal transport for an arbitrary Finsler metric cost [30] (the main advantage of our method being that we never have to evaluate the cost).