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Fisher information regularization schemes for Wasserstein gradient flows

Publication ,  Journal Article
Li, W; Lu, J; Wang, L
Published in: Journal of Computational Physics
September 1, 2020

We propose a variational scheme for computing Wasserstein gradient flows. The scheme builds upon the Jordan–Kinderlehrer–Otto framework with the Benamou-Brenier's dynamic formulation of the quadratic Wasserstein metric and adds a regularization by the Fisher information. This regularization can be derived in terms of energy splitting and is closely related to the Schrödinger bridge problem. It improves the convexity of the variational problem and automatically preserves the non-negativity of the solution. As a result, it allows us to apply sequential quadratic programming to solve the sub-optimization problem. We further save the computational cost by showing that no additional time interpolation is needed in the underlying dynamic formulation of the Wasserstein-2 metric, and therefore, the dimension of the problem is vastly reduced. Several numerical examples, including porous media equation, nonlinear Fokker-Planck equation, aggregation diffusion equation, and Derrida-Lebowitz-Speer-Spohn equation, are provided. These examples demonstrate the simplicity and stableness of the proposed scheme.

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Published In

Journal of Computational Physics

DOI

EISSN

1090-2716

ISSN

0021-9991

Publication Date

September 1, 2020

Volume

416

Related Subject Headings

  • Applied Mathematics
  • 51 Physical sciences
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 02 Physical Sciences
  • 01 Mathematical Sciences
 

Citation

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Li, W., Lu, J., & Wang, L. (2020). Fisher information regularization schemes for Wasserstein gradient flows. Journal of Computational Physics, 416. https://doi.org/10.1016/j.jcp.2020.109449
Li, W., J. Lu, and L. Wang. “Fisher information regularization schemes for Wasserstein gradient flows.” Journal of Computational Physics 416 (September 1, 2020). https://doi.org/10.1016/j.jcp.2020.109449.
Li W, Lu J, Wang L. Fisher information regularization schemes for Wasserstein gradient flows. Journal of Computational Physics. 2020 Sep 1;416.
Li, W., et al. “Fisher information regularization schemes for Wasserstein gradient flows.” Journal of Computational Physics, vol. 416, Sept. 2020. Scopus, doi:10.1016/j.jcp.2020.109449.
Li W, Lu J, Wang L. Fisher information regularization schemes for Wasserstein gradient flows. Journal of Computational Physics. 2020 Sep 1;416.
Journal cover image

Published In

Journal of Computational Physics

DOI

EISSN

1090-2716

ISSN

0021-9991

Publication Date

September 1, 2020

Volume

416

Related Subject Headings

  • Applied Mathematics
  • 51 Physical sciences
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 02 Physical Sciences
  • 01 Mathematical Sciences