Recent Advances in Numerical Solutions for Hamilton-Jacobi PDEs

TL;DR

This paper reviews recent numerical methods for solving Hamilton-Jacobi PDEs, highlighting advances that address high-dimensionality, nonlinearity, and computational efficiency in various applications.

Contribution

It provides a comprehensive overview of recent techniques and emerging directions in numerical solutions for Hamilton-Jacobi PDEs, emphasizing their practical improvements.

Findings

Enhanced algorithms for high-dimensional problems

Improved computational efficiency in nonlinear cases

Identification of promising future research directions

Abstract

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.