Dynamic formulations of L1 optimal transport problems

The purpose of this work is to analyse the relationships between different formulations of the Monge-Kantorovich transport problem, in the case of L 1 cost. In particular, we focus on the famous Benamou and Brenier’s dynamical formulation, adapting it to our cost function. Helped by the solutions of the other formulations, we find a way to solve the dynamical problem, with L 1 cost, and we compare the solution to the one found by Benamou and Brenier, with L 2 cost. To this aim, we also analyse the intermediate cases of L p costs, with 1 < p < 2, and work out a solution procedure based on the nonlinear coupling of a Hamilton-Jacobi equation with a transport equation. We study the qualitative behavior of the solutions to the proposed formulations on simple test cases. When explicit solutions cannot be easily recovered, we use an original numerical approach based on the Finite Volume discretization of the Hamilton-Jacobi and the transport equation and solve the nonlinearities by an ad-hoc adaptation of Newton method.