Primal-dual hybrid gradient algorithms for computing time-implicit Hamilton-Jacobi equations
Year: 2023
Type: preprint
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 $L^1$ Hamiltonians with spatial and time dependence.
Source: arXiv (Cornell University)
Authors: Meng, Tingwei, Hao, Wenbo, Liu, Siting, Osher, Stanley J., Li, Wuchen
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G1523888516, FRG: Collaborative Research: Variationally Stable Neural Networks for Simulation, Learning, and Experimental Design of Complex Physical Systems, G2330613517, G3587679958, G4369523871, G5076589907, G5121863357, G5121974832, G5157887374, G6098521345, G6706517093, G6835302547, G7427201681, G8331043402, G8876996369, RTG: Mathematical Foundation of Data Science at University of South Carolina
OpenAlex ID: W4387355676