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.

翻译：Hamilton-Jacobi（HJ）偏微分方程在物理学、最优控制、博弈论及成像科学等领域具有广泛应用。本研究提出一种基于一阶优化的HJ偏微分方程求解技术，该技术将HJ方程的时间隐式更新重新表述为鞍点问题。我们指出，HJ方程的鞍点公式与最优传输和势平均场博弈（MFG）的原对偶公式具有一致性。这一联系使我们能够拓展MFG方法，设计求解HJ偏微分方程的数值格式。采用原始-对偶混合梯度（PDHG）方法求解鞍点问题，其简洁的结构有助于实现高效的更新计算。值得注意的是，该方法适用于更广泛的哈密顿量类型，包括非光滑及时空依赖的情形。通过一维和二维数值算例（例如具有空间和时间依赖性的二次型及L1哈密顿量）验证了该方法的有效性。