Fisher information regularization schemes for Wasserstein gradient flows

• Published in: Journal of computational physics Vol. 416; p. 109449
• Main Authors: Li, Wuchen, Lu, Jianfeng, Wang, Li
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01-09-2020; Elsevier Science Ltd
• Subjects: Computational physics; Convexity; Fisher information; Fokker-Planck equation; Gradient flow; Interpolation; Optimal transport; Optimization; Porous media; Quadratic programming; Regularization; Schrödinger bridge problem; Time discretization; Time discretization; Gradient flow; Optimal transport; Fisher information; Schrödinger bridge problem
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2020.109449

•Derive Fisher information regularization via energy splitting.•Reduce the dimension of inner dynamic formulation.•With the new regularization, the method is shown to be strictly convex and non-negativity preserving.•The method directly applied to forth order system such as DLSS equation.
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.