Abstract

Optimal Transport (OT) problems arise in a wide range of applications, from physics to economics. Getting numerical approximate solution of these problems is a challenging issue of practical importance. In this work, we investigate the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints. Using Tchakaloff’s theorem, we show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure. Interestingly, for multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which is encouraging to bypass the curse of dimension. Interestingly, the same type of sparsity results also holds in for quantum optimal transport problems stemming from electronic structure calculations. These sparsity results guided the design of new numerical schemes for the resolution of these problems which gave very interesting numerical results in high-dimensional contexts. The end of the talk will be devoted to the remaining open problems related to the mathematical analysis of these schemes.