In [1], Cotar, Friesecke and Klüppelberg rigorously prove that the strongly interacting limit of the exact Hohenberg-Kohn density functional reads as a multi-marginal optimal transport problem with Coulomb cost, confirming results

which had been derived earlier in the physics literature by Seidl [2] and Seidl, Gori-Giorgi and Savin [3]. On the numerical side, computing the solution of this multi-marginal optimal transport problem may be a very challenging task for

systems with a large number of electrons due the curse of dimensionality. The aim of this talk is to present recent results a relaxation of multi-marginal optimal transport problems with

a view to the design of numerical schemes to approximate the exact optimal transport problem. More precisely, the approximate problem considered in this talk consists in relaxing the marginal constraints into a finite number of

moments constraints. Using Tchakhaloff's theorem, it is possible to prove the existence of minimizers of this relaxed problem and characterize them as discrete measures charging a number of points which scales at most linearly with the number of marginals in the problem. This result opens the way to the design of new numerical schemes exploiting the structure of these minimizers, and preliminary numerical results will be presented.

[1] C. Cotar, G. Friesecke, C. Klüppelberg, "Smoothing of transport plans with fixed maginals and rigorous semiclassical limit of the Hohenberg-Kohn functional", 2017, to appear in Arch.Rat.Mech.Analysis.

[2] M. Seidl. Strong-interaction limit of density-functional theory, Phys. Rev. A, 60, 4387-4395 (1999)

[3] M. Seidl, P. Gori-Giorgi, A. Savin. Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities, Phys. Rev. A 75, 042511 1-12 (2007)