## “An energy functional for $L^1$-optimal transport”

ANNULLATO Martedì 3 Marzo 2020, ore 12:30 - Aula seminari 430 - Federico Piazzon (Università di Padova)

Abstract

The optimal transport problem looks for the minimum cost strategy for moving a given mass distribution to a given target configuration. The case of $L^1$ cost has a PDE reformulation, due to Evans and Gangbo, in which one first searches for the transport density $\mu^*$ and a Kantorovich potential $u^*$ that solve a constrained elliptic PDE. Then the transport plan is constructed by moving the mass along $\nabla u^*$ with intensity $\mu^*$.

We introduce a convex energy functional $\mathcal{E}$ whose unique minimizer is precisely the transport density, and for this reason we term it transport energy. The functional $\mathcal{E}$ is designed upon the structure of the Evans-Gangbo PDE and is inspired by the Bouchitté-Buttazzo-Seppecher shape optimization problem.

We study the variational approximation ($\Gamma$-convergence) of $\mathcal{E}$ by a sequence of regularized functionals and the properties of the resulting gradient flows. In particular we characterize $\mu^*$ as the unique global attractor of the gradient flow of $\mathcal{E}$.

We then develop a numerical approximation scheme based on finite elements for the discretization of the $\mu$ and $u$ functional spaces and relying on the minimizing movements strategy for the minimization of the discretized transport energy. We prove the convergence of the scheme and find a Ladyzhenskaya-Babuska-Brezzi inf-sup-like condition that ensures stability and guarantees quadratic convergence of the overall algorithm.