Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation

TL;DR

This paper introduces a numerical method based on constrained overdamped Langevin processes to efficiently solve symmetric multi-marginal optimal transport problems with Coulomb cost, relevant for quantum chemistry applications involving many electrons.

Contribution

The authors develop a novel Langevin-based numerical approach for solving Moment Constrained Optimal Transport relaxations in symmetric multi-marginal problems, exploiting minimizer sparsity and proving global optimality of local solutions.

Findings

Successfully solves MCOT relaxations for systems with up to 100 electrons.

Demonstrates the efficiency of the method through numerical examples.

Leverages sparsity of minimizers to improve computational performance.

Abstract

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.