Fast two-scale methods for eikonal equations. (English) Zbl 1244.49047
Summary: ”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 approaches by using Fast Marching on a coarser scale and Fast Sweeping on a finer scale. We present three new hybrid methods based on this idea and illustrate their properties in several numerical examples with continuous and piecewise-constant speed functions in \(\mathbb R^2\).
MSC:
49L20 | Dynamic programming in optimal control and differential games |
49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
65N06 | Finite difference methods for boundary value problems involving PDEs |
65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
65K05 | Numerical mathematical programming methods |
35F30 | Boundary value problems for nonlinear first-order PDEs |