Virginie Ehrlacher (Ecole des Ponts Paris Tech)

Title: Moment constrained optimal transport problem: application to quantum chemistry
Abstract: This work is motivated by applications in quantum chemistry, for the computation of the electronic structure of molecules. The so-called Density Functional Theory (DFT) is a very powerful framework which enables to carry out such computations. Within this theory, a key role is played by the so-called Levy-Lieb functional, the computation of which remains unaffordable for systems with a large number of electrons. This is why a full zoology of approximate DFT models, relying on the use of various approximations of this Levy-Lieb functional, have been proposed in the chemistry literature. In this talk, a specific focus will be made on one particular DFT model which makes use of the so-called semi-classical limit of the Lévy-Lieb functional, which happens to read as a symmetric multimarginal optimal transport problem with Coulomb cost, the number of marginals being equal to the number of electrons in the system. In this talk, I will present recent results about a new approach for the resolution of multi-marginal optimal transport problems which consists in relaxing the marginal constraints into a finite number of moment 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 independently of the number of electrons. This opens the way towards the design of new numerical schemes which can hopefully circumvent the curse of dimensionality for this problem. Preliminary numerical results will be presented.