Section: New Results

Numerical methods for JKO Gradient Flows

J-D. Benamou, G. Carlier, M. Laborde, G. Peyré, B. Schmitzer, V. Duval

Taking advantage of the Benamou-Brenier dynamic formulation of optimal transport, we propose in [28] , a convex formulation for each step of the JKO scheme for Wasserstein gradient flows which can be attacked by an augmented Lagrangian method which we call the ALG2-JKO scheme. We test the algorithm in particular on the porous medium equation. We also consider a semi implicit variant which enables us to treat nonlocal interactions as well as systems of interacting species. Regarding systems, we can also use the ALG2-JKO scheme for the simulation of crowd motion models with several species.

Figure 12. Evolution of two species where the first one is attracted by the other and the second one is repelled by the first one. Top row: display of ρ1+ρ2. Middle row: display of ρ1. Bottom row: display of ρ2.
IMG/c-autointer-dens-jko-00-copie.png IMG/c-autointer-dens-jko-10-copie.png IMG/c-autointer-dens-jko-20-copie.png IMG/c-autointer-dens-jko-40-copie.png IMG/c-autointer-dens-jko-80-copie.png IMG/c-autointer-dens-jko-100-copie.png
IMG/a-autointer-dens-jko-00-copie.png IMG/a-autointer-dens-jko-10-copie.png IMG/a-autointer-dens-jko-20-copie.png IMG/a-autointer-dens-jko-40-copie.png IMG/a-autointer-dens-jko-80-copie.png IMG/a-autointer-dens-jko-100-copie.png
IMG/b-autointer-dens-jko-00-copie.png IMG/b-autointer-dens-jko-10-copie.png IMG/b-autointer-dens-jko-20-copie.png IMG/b-autointer-dens-jko-40-copie.png IMG/b-autointer-dens-jko-80-copie.png IMG/b-autointer-dens-jko-100-copie.png
t = 0 t = 0 . 1 t = 0 . 2 t = 0 . 4 t = 0 . 8 t = 1

We have also investigated the entropy-regularization of the Wasserstein metric to compute gradient flows [19] , [34] . This entropic regularization trades the usual Wasserstein fidelity term for a Kullback-Leibler divergence term. Adapting first-order proximal methods to this framework, we have developed numerical schemes which dramatically reduce the computational load needed to simulate the evolution of a mass density through a JKO flow. By construction, the entropy regularization yields an additional diffusion effects to the evolution, but we have proved that a careful choice of the regularization parameter with respect to the timestep yields the convergence of the scheme towards the solutions of the continuous PDE.

A novel Lagrangian method using a discretization of the Monge-Ampère operator for JKO has been developed in [13] . Not only convergence of the scheme has been established but also one advantage of this method is that it makes it possible to use a Newton's method .