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Computation of optimal transport with finite volumes

Natale, Andrea; Todeschi, Gabriele (2021), Computation of optimal transport with finite volumes, ESAIM: Mathematical Modelling and Numerical Analysis, 55, 5, p. 1847-1871. 10.1051/m2an/2021041

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Type
Article accepté pour publication ou publié
Date
2021
Journal name
ESAIM: Mathematical Modelling and Numerical Analysis
Volume
55
Number
5
Publisher
EDP Sciences
Pages
1847-1871
Publication identifier
Metadata
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Author(s)
Natale, Andrea
Reliable numerical approximations of dissipative systems [RAPSODI ]
Todeschi, Gabriele
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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.
Subjects / Keywords
Finite volumes; Dynamical optimal transport; Barrier method