Primal dual methods for Wasserstein gradient flows

Understand

Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming.

• Our method proceeds as follows: First, we discretize in time, either via the classical JKO scheme or via a novel Crank-Nicolson type method we introduce.
• Next, we use the Benamou-Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints.
• Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators [Yan 2018].