Numerical Solution of the $L^1$-Optimal Transport Problem on Surfaces

TL;DR

This paper develops a surface finite element method to numerically solve the $L^1$-Optimal Transport Problem on 2D surfaces in 3D, extending previous Euclidean models to Riemannian surfaces and demonstrating improved accuracy and efficiency.

Contribution

It extends the DMK formulation for $L^1$-Optimal Transport to Riemannian surfaces and generalizes the numerical scheme using Surface Finite Element Models.

Findings

The numerical scheme is efficient and robust.

It achieves higher accuracy than existing methods.

Validated on a sphere with promising results.

Abstract

In this article we study the numerical solution of the $L^1$-Optimal Transport Problem on 2D surfaces embedded in $R^3$, via the DMK formulation introduced in [FaccaCardinPutti:2018]. We extend from the Euclidean into the Riemannian setting the DMK model and conjecture the equivalence with the solution Monge-Kantorovich equations, a PDE-based formulation of the $L^1$-Optimal Transport Problem. We generalize the numerical method proposed in [FaccaCardinPutti:2018,FaccaDaneriCardinPutti:2020] to 2D surfaces embedded in $\REAL^3$ using the Surface Finite Element Model approach to approximate the Laplace-Beltrami equation arising from the model. We test the accuracy and efficiency of the proposed numerical scheme, comparing our approximate solution with respect to an exact solution on a 2D sphere. The results show that the numerical scheme is efficient, robust, and more accurate with respect to other numerical schemes presented in the literature for the solution of ls$L^1$-Optimal Transport Problem on 2D surfaces.