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

arXiv: https://arxiv.org/abs/2310.01605 · HTML: https://mlsanity.com/paper/2310.01605/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Oct 2023.

## Built on by (most cited first, 2 of 2)

- [A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations](https://mlsanity.com/paper/2411.06278/) (2024)
- [Neural Networks with Local Converging Inputs for Efficient Options Pricing Models](https://mlsanity.com/paper/2608.02778/) (2026)

## Abstract (arXiv)

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.