Perspectives on Characteristics Based Curse-of-Dimensionality-Free Numerical Approaches for Solving Hamilton–Jacobi Equations

Abstract

This paper extends the considerations of the works (Darbon and Osher, Res Math Sci 3:19, 2016; Chow et al., 2017, arxiv.org/abs/1704.02524) regarding curse-of-dimensionality-free numerical approaches to solve certain types of Hamilton–Jacobi equations arising in optimal control problems, differential games and elsewhere. A rigorous formulation and justification for the extended Hopf–Lax formula of(Chow et al., 2017, arxiv.org/abs/1704.02524) is provided together with novel theoretical and practical discussions including useful recommendations. By using the method of characteristics, the solutions of some problem classes under convexity/concavity conditions on Hamiltonians (in particular, the..