Primal dual Methods for Wasserstein Gradient Flows
In this talk, I will introduce a variational method for nonlinear equations with a gradient flow structure, which arise widely in applications such as porous median flows, material science, animal swarms, and chemotaxis. Our method builds on the JKO framework and a reformulation of the Wasserstein distance into a convex optimization with a linear PDE constraint. As a result, we end up with one nested structure of optimization problem with two time scales, and we adopt a recent primal dual three operator splitting scheme. Thanks to the variational structure, our method has a built-in positivity preserving, entropy decreasing properties, and overcomes stability issue due to the strong nonlinearity and degeneracy. Upon discretization of the PDE constraint, we also show the gamma-convergence of the fully discrete optimization towards the semi-discrete JKO scheme. This is a joint work with Jose Carrillo, Katy Craig, and Chaozhen Wei.