Comptes Rendus
no. 3
Numerical analysis
Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces
[Schémas d'ordre deux en temps pour des flots de gradient dans des espaces métriques géodésiques et de Wasserstein]

Présenté par : Olivier Pironneau

Guillaume Legendre1 ; Gabriel Turinici1, 2
1 Université Paris-Dauphine, PSL Research University, CNRS, UMR 7534, CEREMADE, 75016 Paris, France
2 Institut universitaire de France, France
Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 345-353.

La discrétisation temporelle des flots de gradient dans des espaces métriques utilise des variantes du schéma d'Euler implicite issu du travail séminal de Jordan, Kinderlehrer et Otto [9]. Nous proposons dans cette Note une approche différente, permettant de construire deux schémas numériques d'ordre deux en temps. Dans le cadre d'un espace métrique, nous montrons que les schémas sont bien définis et prouvons la convergence de l'un d'entre eux sous des hypothèses de régularité. Pour le cas particulier d'un flot de gradient Fokker–Planck dans l'espace de Wasserstein, nous obtenons (théoriquement et numériquement) la convergence à l'ordre deux.

The time discretization of gradient flows in metric spaces uses variants of the celebrated implicit Euler-type scheme of Jordan, Kinderlehrer, and Otto [9]. We propose in this Note a different approach, which allows us to construct two second-order in time numerical schemes. In a metric space framework, we show that the schemes are well defined and prove the convergence for one of them under some regularity assumptions. For the particular case of a Fokker–Planck gradient flow in the Wasserstein space, we obtain (theoretically and numerically) the second-order convergence.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.02.001
Guillaume Legendre 1 ; Gabriel Turinici 1, 2

1 Université Paris-Dauphine, PSL Research University, CNRS, UMR 7534, CEREMADE, 75016 Paris, France
2 Institut universitaire de France, France 
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Guillaume Legendre; Gabriel Turinici. Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 345-353. doi : 10.1016/j.crma.2017.02.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.02.001/

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