JKO scheme and gradient flows

in collaboration with J.-D. Benamou, G. Carlier and Q. Merigot

Here are some results obtained in collaboration with J.-D. Benamou, G. Carlier and Q. Merigot. All details can be found in our paper Discretization of functionals involving the Monge-Ampère operator.

Jordan, Kinderlehrer and Otto showed in 1999 how to interpret Fokker-Planck's equation as a gradient flow in the space of probability measures endowed with the Wasserstein distance. This type of formulation has proved to be a powerful tool for studying partial differential equationsinvolving generalized diffusion terms and non-local interaction terms. The construction of Jordan, Kinderlehrer and Otto of the gradient flow of an energy E on the space of probability measures relies on a time-discretization through an implicit Euler scheme. A time step can be written as

where and are probability measures, and is the quadratic Wasserstein distance. Despite the potential applications, there does not exist many numerical methods for this type of scheme, and most of them are restricted to the 1D case. Letting , where is a convex function, a timestep in the JKO scheme can be reformulated as an optimization problem over the space of convex functions. When the energy contains an entropy term, which corresponds to a diffusion term in the underlying PDE, the formulation using involves the Monge-Operator of and is therefore not of the previous form. With G. Carlier, JD Benamou we introduced a space discretization of optimization problems over the space of convex functions and involving the Monge-Ampère operator. In our formulation, an entropy term is transformed into a barrier for the convexity of the discrete functions. This allows us to use Newton's method for the resolution of the discrete problem, which converges in a few iterations. We present numerical applications to the fast diffusion equation, the porous medium equation and to the simulation of crowd motion (see below).