A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows

TL;DR

This paper introduces a splitting scheme for gradient flows with respect to the Kantorovich-Fisher-Rao metric, enabling analysis and numerical solutions of reaction-advection-diffusion equations involving measure spaces with varying masses.

Contribution

It develops a novel splitting JKO scheme for the Kantorovich-Fisher-Rao metric, proving convergence and applicability to reaction-diffusion equations with a constructive numerical approach.

Findings

Proved convergence of the splitting scheme under certain conditions.

Established existence of weak solutions for reaction-advection-diffusion equations.

Demonstrated the scheme's suitability for numerical implementation.

Abstract

In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao distance. Exploiting some inf-convolution structure of the metric we show convergence of the whole process for the standard class of energy functionals under suitable compactness assumptions, and investigate in details the case of internal energies. The interest is double: On the one hand we prove existence of weak solutions for a certain class of reaction-advection-diffusion equations, and on the other hand this process is constructive and well adapted to available numerical solvers.