论文标题
具有成对成本函数的多 - 边界最佳运输的颂歌表征
An ODE characterisation of multi-marginal optimal transport with pairwise cost functions
论文作者
论文摘要
本文的目的是引入一种新的数值方法，以通过成对相互作用的成本解决多界数最佳运输问题。多界限最佳运输的复杂性通常在边际$ m $的数量上成倍扩展。我们引入了一个成本功能的一个参数家族，该函数在原始成本和特殊成本功能之间进行了插值，该功能的复杂性在$ M $中线性地缩放。然后，我们证明可以通过在参数$ε$中求解普通微分方程的原始问题解决方案，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，我们使用明确的Euler和显式的高阶runge-Kutta方案进行一些模拟，以对ode计算解决方案，并因此将多 - 边缘性最佳最佳传输问题计算出来。
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 $ε$, 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.
