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A variational finite volume scheme for Wasserstein gradient flows

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Abstract

We propose a variational finite volume scheme to approximate the solutions to Wasserstein gradient flows. The time discretization is based on an implicit linearization of the Wasserstein distance expressed thanks to Benamou–Brenier formula, whereas space discretization relies on upstream mobility two-point flux approximation finite volumes. The scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level. It can be applied to a wide range of energies, guarantees non-negativity of the discrete solutions as well as decay of the energy. We show that the scheme admits a unique solution whatever the convex energy involved in the continuous problem, and we prove its convergence in the case of the linear Fokker–Planck equation with positive initial density. Numerical illustrations show that it is first order accurate in both time and space, and robust with respect to both the energy and the initial profile.

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Acknowledgements

CC acknowledges the support of the Labex CEMPI (ANR-11-LABX-0007-01). GT acknowledges that this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 754362. We also thank Guillaume Carlier and Quentin Mérigot for fruitful discussions.

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Authors and Affiliations

  1. Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000, Lille, France

    Clément Cancès

  2. Inria, Project Team Mokaplan, Université Paris-Dauphine, PSL Research University, UMR CNRS 7534, Ceremade, France

    Thomas O. Gallouët & Gabriele Todeschi

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  1. Clément Cancès
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Correspondence to Clément Cancès.

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Cancès, C., Gallouët, T.O. & Todeschi, G. A variational finite volume scheme for Wasserstein gradient flows. Numer. Math. 146, 437–480 (2020). https://doi.org/10.1007/s00211-020-01153-9

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