Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?

TL;DR

This paper investigates whether local single-pass methods can solve all stationary Hamilton-Jacobi-Bellman equations, concluding that while generally impossible, they are effective for certain special classes of problems.

Contribution

The paper provides a comprehensive overview of existing fast methods, analyzes their limitations, and demonstrates that local single-pass methods are only applicable to specific problem classes.

Findings

Local single-pass methods cannot solve all Hamilton-Jacobi equations.

These methods are effective for certain special classes of problems.

Numerical tests reveal the limitations and practical usefulness of the methods.

Abstract

The use of local single-pass methods (like, e.g., the Fast Marching method) has become popular in the solution of some Hamilton-Jacobi equations. The prototype of these equations is the eikonal equation, for which the methods can be applied saving CPU time and possibly memory allocation. Then, some natural questions arise: can local single-pass methods solve any Hamilton-Jacobi equation? If not, where the limit should be set? This paper tries to answer these questions. In order to give a complete picture, we present an overview of some fast methods available in literature and we briefly analyze their main features. We also introduce some numerical tools and provide several numerical tests which are intended to exhibit the limitations of the methods. We show that the construction of a local single-pass method for general Hamilton-Jacobi equations is very hard, if not impossible. Nevertheless, some special classes of problems can be actually solved, making local single-pass methods very useful from the practical point of view.