Numerical Solution of the L1-Optimal Transport Problem on Surfaces

In this article we study the numerical solution of the L-Optimal Transport Problem on 2D surfaces embedded in R, via the DMK formulation introduced in [1]. We extend from the Euclidean into the Riemannian setting the DMK model and conjecture the equivalence with the solution MongeKantorovich equations, a PDE-based formulation of the L-Optimal Transport Problem. We generalize the numerical method proposed in [1, 2] to 2D surfaces embedded in R 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 L-Optimal Transport Problem on 2D surfaces.

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