Volume 55, Number 5, September-October 2021
|Page(s)||1847 - 1871|
|Published online||17 September 2021|
Computation of optimal transport with finite volumes
Inria, Project team Rapsodi, Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille, France
2 Inria Paris, Project team Mokaplan, Université Paris-Dauphine, PSL Research University, UMR CNRS 7534-Ceremade, 75016 Paris, France
* Corresponding author: email@example.com
Accepted: 1 August 2021
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.
Mathematics Subject Classification: 65N08 / 35A15 / 65K10 / 49M29 / 90C51
Key words: Dynamical optimal transport / finite volumes / barrier method
© The authors. Published by EDP Sciences, SMAI 2021
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