Time discretizations of Wasserstein-Hamiltonian flows. (English) Zbl 1497.65255
Summary: 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.
MSC:
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
| 49Q22 | Optimal transportation |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
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