Fisher information regularization schemes for Wasserstein gradient flows

Abstract

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 Schr{ö}dinger 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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• Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances
• Hessian metric via transport information geometry
• A dynamic mass transport method for Poisson-Nernst-Planck equations
• Positive and free energy satisfying schemes for diffusion with interaction potentials
• Lagrangian schemes for Wasserstein gradient flows
• A Proximal-Gradient Algorithm for Crystal Surface Evolution
• Parameterized Wasserstein Gradient Flow
• Variational Asymptotic Preserving Scheme for the Vlasov-Poisson-Fokker-Planck System
• Parameter Estimation on Homogeneous Spaces
• Mirror Descent Algorithms for Minimizing Interacting Free Energy