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 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.As written, equation (1) possesses two immediate properties: it preserves the nonnegativity of the solution and conserves total mass. Therefore, in what follows, we will always consider nonnegative initial data with mass one, so that the solution is in the set of probability measures on Ω, P(Ω). The third property of (1) is the dissipation of the energy, which can be seen as follows. Given an energy E : P(Ω) → R ∪ {+∞}, we may formally define its gradient with respect to the quadratic Wasserstein metric W 2 as ∇ W 2 E(ρ) = −∇ · (ρ∇δE) ,

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