Journal article
Primal dual methods for Wasserstein gradient flows
- 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 [107]. 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.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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(Preview, Version of record, 3.8MB, Terms of use)
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- Publisher copy:
- 10.1007/s10208-021-09503-1
Authors
- Publisher:
- Springer
- Journal:
- Foundations of Computational Mathematics More from this journal
- Volume:
- 22
- Issue:
- 2022
- Pages:
- 389–443
- Publication date:
- 2021-03-31
- Acceptance date:
- 2021-02-05
- DOI:
- EISSN:
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1615-3383
- ISSN:
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1615-3375
- Language:
-
English
- Keywords:
- Pubs id:
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1160098
- Local pid:
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pubs:1160098
- Deposit date:
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2021-02-05
Terms of use
- Copyright holder:
- Carrillo et al.
- Copyright date:
- 2021
- Rights statement:
- © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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