Computing the L1 optimal transport density: a FEM approach

The $L^1$ optimal transport density ${\mu }^{\ast }$ is the unique $L^{\mathrm{\infty }}$ solution of the Monge-Kantorovich equations. It has been recently characterized also as the unique minimizer of the $L^1$ -transport energy functional E. In the present work we develop and we prove convergence of a numerical approxi- mation scheme for ${\mu }^{\ast }$ . Our approach relies upon the combination of a FEM- inspired variational approximation of E with a minimization algorithm based on a gradient flow method.