Time discretizations of Wasserstein–Hamiltonian flows

Jianbo Cui, Luca Dieci, Haomin Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

We study discretizations of Hamiltonian systems on the probability density manifold equipped with the L2 -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.

Original languageEnglish
Pages (from-to)1019-1075
Number of pages57
JournalMathematics of Computation
Volume91
Issue number335
DOIs
Publication statusPublished - 14 Mar 2022

Keywords

  • Fisher information
  • optimal transport
  • symplectic schemes
  • Wasserstein—Hamiltonian flow

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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