Fast two-scale methods for Eikonal equations

Fast Marching and Fast Sweeping are the two most commonly used methods for solving the Eikonal equation. Each of these methods performs best on a different set of problems. Fast Sweeping, for example, will outperform Fast Marching on problems where the characteristics are largely straight lines. Fast Marching, on the other hand, is usually more efficient than Fast Sweeping on problems where characteristics frequently change their directions and on domains with complicated geometry. In this paper we explore the possibility of combining the best features of both of these approaches, by using marching on a coarser scale and sweeping on a finer scale. We present three new hybrid methods based on this idea and illustrate their properties on several numerical examples with continuous and piecewise-constant speed functions in $R^2$.