  Many applications require efficient methods for solving continuous shortest
path problems. Such paths can be viewed as characteristics of static
Hamilton-Jacobi equations. Several fast numerical algorithms have been
developed to solve such equations on the whole domain. In this paper we
consider a somewhat different problem, where the solution is needed at one
specific point, so we restrict the computations to a neighborhood of the
characteristic. We explain how heuristic under/over-estimate functions can be
used to obtain a causal domain restriction, significantly decreasing the
computational work without sacrificing convergence under mesh refinement. The
discussed techniques are inspired by an alternative version of the classical A*
algorithm on graphs. We illustrate the advantages of our approach on continuous
isotropic examples in 2D and 3D. We compare its efficiency and accuracy to
previous domain restriction techniques. We also analyze the behavior of errors
under the grid refinement and show how Lagrangian (Pontryagin's Maximum
Principle-based) computations can be used to enhance our method.
