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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]

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. [ABSTRACT FROM AUTHOR]

We study the geodesic problem on the group of diffeomorphism of a domain M ⊂ R d , equipped with the H (div) metric. The geodesic equations coincide with the Camassa–Holm equation when d = 1 , and represent one of its possible multi-dimensional generalizations when d > 1 . We propose a relaxation à la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M. We use this relaxation to prove that smooth H (div) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure field associated to solutions of our relaxation. Finally, we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa–Holm and incompressible Euler solutions. [ABSTRACT FROM AUTHOR]

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]

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. [ABSTRACT FROM AUTHOR]

We investigate gradient flows of some homogeneous functionals in R N , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong self-interaction (super critical case), the functional possesses a cone of negative energy. It is immediate to see that solutions with negative energy at some time become singular in finite time, meaning that a subset of particles concentrate at a single point. Here, we establish that all solutions become singular in finite time, in the super critical case, for the class of functionals under consideration. The paper is completed with numerical simulations illustrating the striking non linear dynamics when initial data have positive energy. [ABSTRACT FROM AUTHOR]

We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D. [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]