2023/10/02 by Tingwei Meng, Wenbo Hao, Meng, Tingwei +7
Mathematics · Physics and Astronomy · #FOS: Mathematics #Mathematical Biology Tumor Growth #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2310.01605
openalex publication_date 2023/10/02 · openalex created_date 2023/10/05 · openalex updated_date 2026/08/01
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 L1 Hamiltonians with spatial and time dependence.