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Mathematics > Numerical Analysis

arXiv:2310.01605 (math)
[Submitted on 2 Oct 2023]

Title:Primal-dual hybrid gradient algorithms for computing time-implicit Hamilton-Jacobi equations

Authors:Tingwei Meng, Wenbo Hao, Siting Liu, Stanley J. Osher, Wuchen Li
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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.
Subjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
Cite as: arXiv:2310.01605 [math.NA]
  (or arXiv:2310.01605v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2310.01605

Submission history

From: Tingwei Meng [view email]
[v1] Mon, 2 Oct 2023 19:57:22 UTC (505 KB)
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