Primal Dual Methods for Wasserstein Gradient Flows

José A Carrillo,Chaozhen Wei,Katy Craig,Li Wang

doi:10.1007/s10208-021-09503-1

Abstract

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 in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

Highlights

Gradient flow methods are classical techniques for the analysis and numerical simulation of partial differential equations
We consider a dynamical reformulation of these minimization problems, stemming from Benamou and Brenier’s dynamic characterization of the Wasserstein metric, by which the problem becomes the minimization of a strictly convex integral functional subject to a linear PDE constraint
In Sect. 4.2.2, we provide numerical examples comparing the above method to the classical JKO scheme from Problem 1, illustrating that it achieves a higher-order rate of convergence in practice, in spite of the fact that that it lacks the energy decay property of Problem 1

Summary

Introduction

Gradient flow methods are classical techniques for the analysis and numerical simulation of partial differential equations Such methods were exclusively based on gradient flows arising from a Hilbert space structure, L2(Rd ), but since the work of Jordan, Kinderlehrer, and Otto in the late 90’s [75,93,94], interest has emerged in a range of nonlinear, nonlocal partial differential equations that are gradient flows in the Wasserstein metric,. When Ω = Rd , we consider no-flux boundary conditions Equations of this form arise in a number of physical and biological applications, including models in granular media [12,45,46,102], material science [71], and biological swarming [6,39,77]. When the interaction potential W is given by a repulsive–attractive Morse or power-law potential,

Objectives

Methods

Results

Conclusion