Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation

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 number marginal laws is equal electrons system, which can be very large relevant applications. In this work, we design numerical method, built upon constrained overdamped Langevin processes solve Moment Constrained Optimal Transport (MCOT) relaxations (introduced [A. Alfonsi, R. Coyaud, V. Ehrlacher and D. Lombardi, Approximation problems moments constraints, Math. Comp. 90 (2021) 689–737; C. Villani, Transport: Old New (Springer Science & Business Media, 2008)]) cost. Some minimizers such written as discrete measures charging low points belonging space whose dimension, symmetrical case, scales linearly laws. We leverage sparsity those method prove that there no strict local minimizer resulting problem. illustrate performance proposed by examples solves MCOT 3D systems up 100 electrons.