Computation of Optimal Transport with Finite Volumes

Andrea Natale (1) , Gabriele Todeschi (2)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales

We construct Two-Point Flux Approximation (TPFA) finite volume schemes to solve the quadratic optimal transport problem in its dynamic form, namely the problem originally introduced by Benamou and Brenier. We show numerically that these type of discretizations are prone to form instabilities in their more natural implementation, and we propose a variation based on nested meshes in order to overcome these issues. Despite the lack of strict convexity of the problem, we also derive quantitative estimates on the convergence of the method, at least for the discrete potential and the discrete cost. Finally, we introduce a strategy based on the barrier method to solve the discrete optimization problem.

Domaines: Analyse numérique [math.NA]

Andrea Natale, Gabriele Todeschi. Computation of Optimal Transport with Finite Volumes. 2020. ⟨hal-03348217v1⟩