Section: New Results

JKO gradient flow numerics

• Benamou, Carlier, Merigot (Univ. of Grenoble, CNRS) , Oudet (Univ. of Grenoble)q

  A large class of non-linear continuity equations with confinement and/or possibly non local interaction potential can be considered as semi discrete gradient flows with respect to the Euclidean Wassertein distance. The numerical resolution of such problem in dimension 2 and higher is open. Our approach is based on two remarks : the reformulation of the optimization problem in terms of Brenier potential seems to behave better. This introduces a Monge-Ampère operator in the cost functional which needs a monotone discretization in order to preserve the convexity at the discrete level. The first numerical results are very encouraging.

  Figure 5. One step of Wasserstein JKO gradient flow for the classical entropy (our numerical method) compared to traditional Finite Difference of the heat equation. Left the initial heat profile, right the heat profile after one time step for both methods.

• Benamou, Carlier, Agueh (Univ. of Victoria) Splitting methods for kinetic equations, we try to use one JKO step to deal with the non-linear velocity advection part of kinetic equations [31] . This seems to be relevant to granular media equation [16] , and also may offer a completely new method for Liouville equations arising from Geometrical Optics [19] .