  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.
