A fast sweeping method for Eikonal equations

Abstract

Abstract. In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of O(N ) for N grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

“…Min and Gibou also used the idea of Russo and Smereka with slight modifications in the context of adaptive mesh refinement [90], and Min pointed out that it is advantageous in terms of speed and memory to replace the traditional Runge-Kutta scheme in time with a Gauss-Seidel iteration of the forward Euler scheme [88]. Finally, we mention that other techniques can be used to reinitialize φ as a distance function [125,124,149,166,148,31,147,53], each with their pros and cons. We refer the interested readers to the book by Osher and Fedkiw [101] as well to the book by Sethian [127] for more details on the level-set method.…”

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

“…It is shown in [40] that the complexity of this algorithm is O(f max /f min N ) in order to achieve an accuracy that is independent of the variation of f (x). In the fast sweeping method [5,55,47,54,27,28,53,38,39,52], Gauss-Seidel iterations with alternating orderings is combined with upwind finite differences. In contrast to the fast marching method, the fast sweeping method follows the causality along characteristics in a parallel way; i.e., all characteristics are divided into a finite number of groups according to their directions and each Gauss-Seidel iteration with a specific sweeping ordering covers a group of characteristics simultaneously; no heap-sort is needed.…”

First-arrival traveltime tomography for anisotropic media using the adjoint-state method

“…The solution of equation 2 can be computed efficiently by fast sweeping methods (Zhao, 2005). Specifically, we use an iterative fast sweeping-based algorithm to solve the acoustic anisotropic eikonal equation.…”