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:本文的目的是采用一种新的数字方法,用双向互动成本解决多边最佳运输问题。多边最佳运输的复杂性一般以边际数成指数比例。我们引入一个成本函数参数组合,将问题的复杂度线性地以百万美元计算在原始和特殊成本函数之间,从而将这一问题的复杂度线以线性计算成百万美元。然后我们表明,可以通过解决参数$\epsilon$中的普通差分方程来找到最初问题的解决方案,该方程的初始条件与上述特殊成本函数的解决方案相对应;然后我们提出一些模拟,使用明确的欧勒和明确的更高顺序Runge-Kutta方案来计算对ODE的解决方案,并由此产生多边际最佳运输问题。