L. Nenna: From (optimal) fat plans to optimal maps for the multi-marginal optimal transport with Coulomb cost
Abstract: In this talk I will present the multi-marginals (Monge-Kantorovich) optimal transport problem (with a generic cost function) and a numerical method to solve this kind of problem based on the discretization of the entropic functional (also known as Kullback-Leibler distance). Then, I will introduce the Density Functional Theory (DFT) which is an approximate computational method for solving the many-electron Schrödinger equation at a more reasonable cost.
One scenario of interest for the DFT is when the repulsion between electrons largely dominates over the kinetic energy. In this case the problem can be reformulated as a multi-marginal OT problem with the Coulomb cost. I will show that the entropic functional, firstly used as a numerical trick, has actually an interesting physical meaning. Finally, I will present some numerical results highlighting that the solution of the OT problem with Coulomb cost can be either a "fat" transport plan or a transport map.