Recent Advances in Numerical Solutions for Hamilton-Jacobi PDEs

Hamilton-Jacobi partial differential equations (HJ PDEs) play a central role in many applications such as economics, physics, and engineering. These equations describe the evolution of a value function that encodes valuable information about the system, such as action, cost, or level sets of a dynamic process. Their importance lies in their ability to model diverse phenomena, ranging from the propagation of fronts in computational physics to optimal decision-making in control systems. This paper provides a review of some recent advances in numerical methods to address challenges such as high dimensionality, nonlinearity, and computational efficiency. By examining these developments, this paper sheds light on important techniques and emerging directions in the numerical solution of HJ PDEs.