We consider the numerical solution of the optimal transport problem between densities that are supported on sets of unequal dimension. Recent work by McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type equation. We provide a new level set framework for interpreting this non-linear PDE. We also propose a novel discretisation that combines carefully constructed monotone finite difference schemes with a variable-support discrete version of the Dirac delta function. The resulting method is consistent and monotone. These new techniques are described and implemented in the setting of 1D to 2D transport, but can easily be generalised to higher dimensions. Several challenging computational tests validate the new numerical method.

翻译：本文研究在支撑集维数不等的密度之间求解最优传输问题的数值方法。McCann与Pass近期的工作将该问题重构为一种非局部Monge-Ampère型方程。我们为这一非线性偏微分方程提出了一种新的水平集框架，并设计了一种新颖的离散化方案，该方案将精心构造的单调有限差分格式与支持集可变的离散Dirac delta函数相结合。由此得到的方法具有相容性与单调性。这些新技术在一维到二维传输场景中进行了描述与实现，但可轻松推广至更高维度。多项具有挑战性的数值测试验证了该新方法的有效性。