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We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. We propose several possible definitions for such an operation, and we prove convergence of the resulting scheme to the limit partial differential equation (PDE), in the case of the Fokker-Planck equation. For a specific choice of extrapolation we also prove a more general result, that is convergence towards Evolutional Variational Inequality flows. Finally, we propose a variational finite volume discretization of the scheme which numerically achieves second order accuracy in both space and time. [ABSTRACT FROM AUTHOR]

On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we propose a simpler approach based on the relaxation of the variational problem on the path space. We explore two different numerical approaches, one based on multimarginal optimal transport and entropic regularization and the other based on semi-discrete optimal transport., Author(s): Jean-David Benamou [sup.1] [sup.2] , Thomas O. Gallouët [sup.1] [sup.2] , François-Xavier Vialard [sup.3] Author Affiliations: (Aff1) grid.5328.c, 0000 0001 2186 3954, Project Team Mokaplan, INRIA, , 2 Rue [...]

The Wasserstein gradient flow structure of the partial differential equation system governing multiphase flows in porous media was recently highlighted in Cancès et al. [ Anal. PDE 10 (8), 1845–1876]. The model can thus be approximated by means of the minimising movement (or JKO after Jordan, Kinderlehrer and Otto [ SIAM J. Math. Anal. 29 (1), 1–17]) scheme that we solve thanks to the ALG2-JKO scheme proposed in Benamou et al. [ ESAIM Proc. Surv. 57 , 1–17]. The numerical results are compared to a classical upstream mobility finite volume scheme, for which strong stability properties can be established. [ABSTRACT FROM AUTHOR]

In this article we set up a splitting variant of the Jordan–Kinderlehrer–Otto scheme in order to handle gradient flows with respect to the Kantorovich–Fisher–Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge–Kantorovich distance and then for the Hellinger/Fisher–Rao distance. Exploiting some inf-convolution structure of the metric we show convergence of the whole process for the standard class of energy functionals under suitable compactness assumptions and investigate in details the case of internal energies. The interest is double: On the one hand we prove existence of weak solutions for a certain class of reaction-advection-diffusion equations, and on the other hand this process is constructive and well adapted to available numerical solvers. [ABSTRACT FROM AUTHOR]

Given a smooth nonfocal compact Riemannian manifold, we show that the so-called Ma--Trudinger--Wang condition implies the convexity of injectivity domains. This improves a previous result by Loeper and Villani. [ABSTRACT FROM AUTHOR]