Time discretizations of Wasserstein–Hamiltonian flows

Cui, Jianbo; Dieci, Luca; Zhou, Haomin

doi:10.1090/mcom/3726

Time discretizations of Wasserstein–Hamiltonian flows
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by Jianbo Cui, Luca Dieci and Haomin Zhou;

Math. Comp. 91 (2022), 1019-1075

DOI: https://doi.org/10.1090/mcom/3726

Published electronically: March 14, 2022

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Abstract:

We study discretizations of Hamiltonian systems on the probability density manifold equipped with the $L^2$-Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems on a graph (lattice) with different weights are derived, which can be viewed as spatial discretizations of the original Hamiltonian systems. We prove consistency of these discretizations. Furthermore, by regularizing the system using the Fisher information, we deduce an explicit lower bound for the density function, which guarantees that symplectic schemes can be used to discretize in time. Moreover, we show desirable long time behavior of these symplectic schemes, and demonstrate their performance on several numerical examples. Finally, we compare the present approach with the standard viscosity methodology.

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