Primal Dual Methods for Wasserstein Gradient Flows

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

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)389-443
Number of pages55
JournalFoundations of Computational Mathematics
Issue number2
StatePublished - Mar 31 2021

Bibliographical note

Funding Information:
The authors would like to thank Ming Yan for fruitful discussions on primal dual methods and preliminary code for the three operator splitting algorithm. They would like to thank Wuchen Li for many helpful conversations. Finally, they would like to thank the anonymous reviewers for their useful observations and suggestions, which greatly improved this work. JAC was supported the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 883363). JAC was also partially supported by the EPSRC Grant Number EP/P031587/1. KC was supported by NSF DMS-1811012 and a Hellman Faculty Fellowship. LW and CW are partially supported by NSF DMS-1620135, 1903420, 1846854.

Publisher Copyright:
© 2021, The Author(s).


  • Gradient flows
  • Minimizing movements
  • Optimal transport
  • Optimization schemes
  • Primal dual methods
  • Steepest descent schemes


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