Causal Domain Restriction for Eikonal Equations

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 two and three dimensions. We compare its efficiency and accuracy to previous domain restriction techniques. We also analyze the behavior of errors under the grid refinemen...