CAN LOCAL SINGLE-PASS METHODS SOLVE ANY STATIONARY HAMILTON-JACOBI-BELLMAN EQUATION?

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 questions naturally arise: Can local single-pass methods solve any Hamilton-Jacobi equation? If not, where should the limit 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 the literature and 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 actually be solved, making local single-pass methods very useful from a practical point of view.