Contributions to the numerical study of Optimal Transport

Optimal Transport and the Monge-Kantorovich problem recently revived both from a theoretical and a practical standpoint. Optimal mass transfer problems have many connections with PDE systems arising in meteorology, electrodynamics, crowd motion, optimal design.

After a brief introduction on the Monge-Kantorovich problem, a lagrangian class of numerical methods is presented to approximate its solution. It is compared in terms of formulation and computational costs with Benamou-Brenier and AHT methods.

Then, a numerical study on how the Wasserstein distance could be used to study low order representations of systems of PDEs is detailed and some numerical experiments discussed. The method proposed is effective when the solutions are featured by the transport of concentrated structures.

In the last part, a still ongoing work is presented on unbalanced optimal transport problems. Several mass sources models are described. In the end, an equivalence between mass sources models and metrics of the space is presented, aiming at modeling obstacles and constraints.