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 which 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.

翻译：Hamilton-Jacobi偏微分方程（HJ PDEs）在经济学、物理学和工程学等众多应用领域中扮演着核心角色。这些方程描述了价值函数的演化过程，该函数编码了系统的重要信息，如动态过程的作用量、代价或水平集。其重要性在于能够建模从计算物理中的波前传播到控制系统中最优决策等多样化现象。本文综述了近年来针对高维性、非线性和计算效率等挑战的数值方法进展。通过考察这些发展，本文揭示了HJ PDEs数值求解中的重要技术与新兴方向。