The purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one parameter family of cost functions that interpolates between the original and a special cost function for which the problem's complexity scales linearly in $m$. We then show that the solution to the original problem can be recovered by solving an ordinary differential equation in the parameter $\epsilon$, whose initial condition corresponds to the solution for the special cost function mentioned above; we then present some simulations, using both explicit Euler and explicit higher order Runge-Kutta schemes to compute solutions to the ODE, and, as a result, the multi-marginal optimal transport problem.

翻译：本文旨在介绍一种新的数值方法，用于求解具有成对交互成本的多边缘最优输运问题。多边缘最优输运的复杂度通常随边缘数量$m$呈指数增长。我们引入一个含单参数的成本函数族，它在原始成本函数与一种特殊成本函数之间进行插值，其中对于后者，问题的复杂度随$m$线性增长。随后，我们证明原始问题的解可通过求解关于参数$\epsilon$的常微分方程恢复，该方程的初始条件对应于上述特殊成本函数的解；最后，我们给出若干数值模拟，分别使用显式欧拉法和显式高阶龙格-库塔格式计算该常微分方程的解，从而得到多边缘最优输运问题的解。