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.
Bibliographical noteFunding Information:
WL was partially supported by AFOSR MURI FA9550-18-1-0502 . JL was partially supported by NSF under grant DMS-1454939 . LW was partially supported by NSF grant DMS-1903420 and NSF CAREER grant DMS-1846854 . The authors are grateful to the support from KI-Net ( NSF grant RNMS-1107444 ) and UMN-Math Visitors Program to facilitate the collaboration.
© 2020 Elsevier Inc.
- Fisher information
- Gradient flow
- Optimal transport
- Schrödinger bridge problem
- Time discretization