A variational finite volume scheme for Wasserstein gradient flows

Clément Cancès 1 Thomas Gallouët 2 Gabriele Todeschi 2
1 RAPSODI - Reliable numerical approximations of dissipative systems
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
Inria de Paris, CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
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. Our 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. Our scheme 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 our 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.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

Cited literature [57 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02189050
Contributor : Clément Cancès <>
Submitted on : Friday, July 19, 2019 - 12:14:44 AM
Last modification on : Friday, February 14, 2020 - 1:13:45 AM

File

CGT_LJKO.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02189050, version 1

Collections

Citation

Clément Cancès, Thomas Gallouët, Gabriele Todeschi. A variational finite volume scheme for Wasserstein gradient flows. 2019. ⟨hal-02189050⟩

Share

Metrics

Record views

102

Files downloads

121