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.

翻译：高密度功能理论(DFT)中功能性高密度功能理论(Levy-Lieb)功能的严格相关电子(SCE)限制引发了具有库伦成本的对称多边最佳运输问题,其边际法律的数量相当于系统中的电子数量,在相关应用中这种数量可能非常大。在这项工作中,我们设计了一种数字方法,它建立在受限制的高压兰格文程序之上,以解决在高密度功能理论(DFT)中功能性能最佳运输(A. Alfonsi、R. Coyaud、V. Ehrlacher和D. Lombardi, Math. comp. 90、2021、689-737 等值的多边际最佳运输问题与库伦巴迪费用相当。在这种自由度规则中,我们利用这些最小化的最小化器在设计中,以100度的最佳性能方法显示一个数字-数字-数字-数字-数字-数字-数字-数字-任何我们用一个数字-数字-数字-数字方法证明任何最小化的方法。