TIME DISCRETIZATIONS OF WASSERSTEIN-HAMILTONIAN FLOWS

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.

Automated summary

This paper studies the discretizations of Hamiltonian systems on the probability density manifold with the L-2-Wasserstein metric. Based on discrete optimal transport theory, Hamiltonian systems on a graph with different weights are derived, which serve as spatial discretizations of the original systems. The consistency of these discretizations is proven. Moreover, by regularizing the system and obtaining an explicit lower bound for the density function, the use of symplectic schemes for time discretization is guaranteed. Desirable long time behavior of these symplectic schemes is shown and their performance is demonstrated through numerical examples. Finally, the present approach is compared with the standard viscosity methodology.