Primal-dual hybrid gradient algorithms for computing time-implicit Hamilton-Jacobi equations
Abstract
Hamilton-Jacobi (HJ) partial differential equations (PDEs) have diverse applications spanning physics, optimal control, game theory, and imaging sciences. This research introduces a first-order optimization-based technique for HJ PDEs, which formulates the time-implicit update of HJ PDEs as saddle point problems. We remark that the saddle point formulation for HJ equations is aligned with the primal-dual formulation of optimal transport and potential mean-field games (MFGs). This connection enables us to extend MFG techniques and design numerical schemes for solving HJ PDEs. We employ the primal-dual hybrid gradient (PDHG) method to solve the saddle point problems, benefiting from the simple structures that enable fast computations in updates. Remarkably, the method caters to a broader range of Hamiltonians, encompassing non-smooth and spatiotemporally dependent cases. The approach's effectiveness is verified through various numerical examples in both one-dimensional and two-dimensional examples, such as quadratic and Hamiltonians with spatial and time dependence.
Cite
@article{arxiv.2310.01605,
title = {Primal-dual hybrid gradient algorithms for computing time-implicit Hamilton-Jacobi equations},
author = {Tingwei Meng and Wenbo Hao and Siting Liu and Stanley J. Osher and Wuchen Li},
journal= {arXiv preprint arXiv:2310.01605},
year = {2023}
}