## Time discretizations of Wasserstein–Hamiltonian flows

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Jianbo Cui, Luca Dieci and Haomin Zhou
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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.## References

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## Additional Information

**Jianbo Cui**- Affiliation: School of Mathematics, Georgia Tech, Atlanta, Georgia 30332
- Address at time of publication: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
- Email: jianbo.cui@polyu.edu.hk
**Luca Dieci**- Affiliation: School of Mathematics, Georgia Tech, Atlanta, Georgia 30332
- MR Author ID: 247339
- Email: dieci@math.gatech.edu
**Haomin Zhou**- Affiliation: School of Mathematics, Georgia Tech, Atlanta, Georgia 30332
- MR Author ID: 643741
- Email: hmzhou@math.gatech.edu
- Received by editor(s): June 11, 2020
- Received by editor(s) in revised form: September 9, 2021
- Published electronically: March 14, 2022
- Additional Notes: The research was partially supported by Georgia Tech Mathematics Application Portal (GT-MAP) and by research grants NSF DMS-1620345. DMS-1830225, and ONR N00014-18-1-2852. The research of the first author was partially supported by start-up funds (P0039016) from Hong Kong Polytechnic University and the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics.

The first author is the corresponding author. - © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 1019-1075 - MSC (2020): Primary 65P10; Secondary 35R02, 58B20, 65M12
- DOI: https://doi.org/10.1090/mcom/3726
- MathSciNet review: 4405488