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

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 p...

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Veröffentlicht in:Research in the mathematical sciences Jg. 12; H. 2; S. 37
Hauptverfasser: Meng, Tingwei, Hao, Wenbo, Liu, Siting, Osher, Stanley J., Li, Wuchen
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cham Springer International Publishing 01.06.2025
Springer Nature B.V
Schlagworte:
Algorithms
Applications of Mathematics
Computational Mathematics and Numerical Analysis
Computing time
Conservation laws
Game theory
Hamilton-Jacobi equation
Lagrange multiplier
Mathematics
Mathematics and Statistics
Optimal control
Optimization
Partial differential equations
Saddle points
Time dependence
Time-implicit scheme
Hamilton–Jacobi PDEs
65M06
Primal–dual hybrid gradient algorithms
Optimization for PDEs
ISSN:2522-0144, 2197-9847
Online-Zugang:Volltext
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Zusammenfassung: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 some 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.
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ISSN:2522-0144
2197-9847
DOI:10.1007/s40687-025-00519-5