A Volumetric Approach to Monge's Optimal Transport on Surfaces

TL;DR

This paper introduces a volumetric method for solving Monge's Optimal Transport problem on surfaces by extending it to a thin tubular region, simplifying numerical computation while maintaining accuracy.

Contribution

It presents a novel volumetric formulation for optimal transport on surfaces, enabling easier numerical discretization and solution on complex geometries.

Findings

Effective for surfaces like the sphere and torus

Uses simple numerical methods on Cartesian grids

Ensures consistency with original surface problem

Abstract

We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in $\mathbb{R}^3$, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the original problem on the surface, we define a new Optimal Transport problem on a thin tubular region, $T_{\epsilon}$, adjacent to the surface. This extension offers enhanced flexibility and simplicity for numerical discretization on Cartesian grids. The Optimal Transport mapping and potential function computed on $T_{\epsilon}$ are consistent with the original problem on surfaces. We demonstrate that, with the proposed volumetric approach, it is possible to use simple and straightforward numerical methods to solve Optimal Transport for $\Gamma = \mathbb{S}^2$ and the $2$-torus.