Numerical Solution of the $L^1$-Optimal Transport Problem on Surfaces
Abstract
In this article we study the numerical solution of the -Optimal Transport Problem on 2D surfaces embedded in , 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 -Optimal Transport Problem. We generalize the numerical method proposed in [FaccaCardinPutti:2018,FaccaDaneriCardinPutti:2020] to 2D surfaces embedded in 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-Optimal Transport Problem on 2D surfaces.
Cite
@article{arxiv.2106.06479,
title = {Numerical Solution of the $L^1$-Optimal Transport Problem on Surfaces},
author = {Luca Berti and Enrico Facca and Mario Putti},
journal= {arXiv preprint arXiv:2106.06479},
year = {2024}
}