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

Luca Berti, Enrico Facca, Mario Putti

TL;DR

This work tackles the numerical solution of the $L^{1}$-optimal transport problem on 2D surfaces by extending the Dynamic Monge-Kantorovich (DMK) formulation to Riemannian manifolds and solving it with Surface Finite Element Methods. By discretizing in space with SFEM and in time via forward Euler, the authors obtain time-converged approximations to the Monge-Kantorovich equations and Beckmann problem, and they demonstrate superior accuracy and convergence compared to the Earth Mover's Distance ADMM approach on a sphere. The key contributions include a geometry-aware DMK discretization on surfaces, a stable MOL scheme with inf-sup stabilization, and rigorous numerical validation against closed-form MK solutions, including explicit computation of the Wasserstein-1 distance $W_{1}$. The results indicate that the proposed DMK-based method is efficient, robust, and capable of achieving higher accuracy, providing a promising tool for OT on curved manifolds with potential extensions to multilevel and implicit-time schemes.

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.

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

TL;DR

This work tackles the numerical solution of the -optimal transport problem on 2D surfaces by extending the Dynamic Monge-Kantorovich (DMK) formulation to Riemannian manifolds and solving it with Surface Finite Element Methods. By discretizing in space with SFEM and in time via forward Euler, the authors obtain time-converged approximations to the Monge-Kantorovich equations and Beckmann problem, and they demonstrate superior accuracy and convergence compared to the Earth Mover's Distance ADMM approach on a sphere. The key contributions include a geometry-aware DMK discretization on surfaces, a stable MOL scheme with inf-sup stabilization, and rigorous numerical validation against closed-form MK solutions, including explicit computation of the Wasserstein-1 distance . The results indicate that the proposed DMK-based method is efficient, robust, and capable of achieving higher accuracy, providing a promising tool for OT on curved manifolds with potential extensions to multilevel and implicit-time schemes.

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.

Paper Structure

This paper contains 17 sections, 1 theorem, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The $L^{1}$ optimal transportation problem
  3. The Kantorovich problem
  4. The MK equations
  5. The Beckmann Problem
  6. The Dynamic Monge-Kantorovich equations
  7. Extension to manifolds
  8. Numerical Discretization
  9. Spatial discretization
  10. Time discretization
  11. Approximate solutions of the MK equations and Beckmann problem
  12. Numerical experiments
  13. Comparison with closed-form solutions
  14. Description of the test case
  15. Experimental results
  16. ...and 2 more sections

Key Result

Proposition 1

The OT density $\mu^*$ is the unique minimizer of $\mathcal{L}$ and the corresponding infimum value is exactly the Wasserstein-1 distance between $f^{+}$ and $f^{-}$: $\mathcal{L}(\mu^*)=W_{1}(f^{+},f^{-})$. Moreover, for any $T>0$ for which a solution pair $(\mu(t,\cdot), u(t,\cdot)$ of prob:dmk e

Figures (3)

  • Figure 1: Spatial distribution of the $L^{2}$-projection of $f^{+} -f^{-}$ on the mesh $\mathcal{T}$ with $554$ nodes and $1104$ triangles (left). Spatial distribution of the density $\mu^*$ together with the contour lines of the supports of $f^{+}$ and $f^{-}$ (right).
  • Figure 2: Experimental convergence towards the MK solution. Left column: time ($t$) evolution of $\operatorname{var}(\mu_{ h }(t))$, which measures convergence towards equilibrium, and of $\operatorname{err}(v_{ h }(t))$ and $\operatorname{err}_{W_{1}}(\mu_{ h }(t))$, which measure the accuracy of the spatial approximation. The results are obtained on successive uniform refinements of a an initial mesh with 554 nodes and 1124 triangles. Right column: values of the four simulation metrics as a function of computational time ($t_{CPU}$, seconds). The simulations were conducted on a first-generation 2.2GHz Intel-I5 (1-core) laptop computer.
  • Figure 3: Spatial convergence of DMK and EMDADMM on the relative Beckmann error $\operatorname{err}_{BP}(v_{ h }^*)$ (left) and on the Wasserstein-1 distance $\operatorname{err}_{W_{1}}(\mu_{ h }^*)$ (right). The errors are calculated on four mesh refinements and the corresponding experimental spatial convergence rate with respect to mesh parameter $h$, evaluated by linear approximation, is reported in the plot legend.

Theorems & Definitions (3)

  • Proposition 1
  • Remark 1
  • Definition 1: Tangential Gradient