  The Strictly Correlated Electrons (SCE) limit of the Levy-Lieb functional in
Density Functional Theory (DFT) gives rise to a symmetric multi-marginal
optimal transport problem with Coulomb cost, where the number of marginal laws
is equal to the number of electrons in the system, which can be very large in
relevant applications. In this work, we design a numerical method, built upon
constrained overdamped Langevin processes to solve Moment Constrained Optimal
Transport (MCOT) relaxations (introduced in A. Alfonsi, R. Coyaud, V. Ehrlacher
and D. Lombardi, Math. Comp. 90, 2021, 689--737) of symmetric multi-marginal
optimal transport problems with Coulomb cost. Some minimizers of such
relaxations can be written as discrete measures charging a low number of points
belonging to a space whose dimension, in the symmetrical case, scales linearly
with the number of marginal laws. We leverage the sparsity of those minimizers
in the design of the numerical method and prove that any local minimizer to the
resulting problem is actually a \emph{global} one. We illustrate the
performance of the proposed method by numerical examples which solves MCOT
relaxations of 3D systems with up to 100 electrons.
