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Journal of Convex Analysis 31 (2024), No. 2, 359--378
Copyright Heldermann Verlag 2024

Wasserstein Gradient Flow of the Fisher Information from a Non-Smooth Convex Minimization Viewpoint

Guillaume Carlier
CEREMADE, UMR CNRS 7534, Université Paris Dauphine, Paris, France

Jean-David Benamou
MOKAPLAN INRIA, Université Paris Dauphine, Paris, France

Daniel Matthes
Dept. of Mathematics, School of CIT, Technische Universität, München, Germany

Motivated by the Derrida-Lebowitz-Speer-Spohn (DLSS) quantum drift diffusion equation, which is the Wasserstein gradient flow of the Fisher information, we study in details solutions of the corresponding implicit Euler scheme. We also take advantage of the convex (but non-smooth) nature of the corresponding variational problem to propose a numerical method based on the Chambolle-Pock primal-dual algorithm.

Keywords: DLSS equation, Wasserstein distance, Fisher information, convex duality, Chambolle-Pock algorithm.

MSC: 49Q22, 65K10.

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