Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?
For researchers in numerical methods for PDEs, this clarifies the fundamental limitations of local single-pass methods, showing they cannot be universally applied beyond specific cases like the eikonal equation.
This paper investigates whether local single-pass methods (e.g., Fast Marching) can solve general stationary Hamilton-Jacobi-Bellman equations, concluding that constructing such methods for general equations is very hard if not impossible, though they remain useful for special problem classes.
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.