Some Improvements for the Fast Sweeping Method

Abstract

In this paper, we outline two improvements to the fast sweeping method to improve the speed of the method in general and more specifically in cases where the speed is changing rapidly. The conventional wisdom is that fast sweeping works best when the speed changes slowly, and fast marching is the algorithm of choice when the speed changes rapidly. The goal here is to achieve run times for the fast sweeping method that are at least as fast, or faster, than competitive methods, e.g. fast marching, in the case where the speed is changing rapidly. The first improvement, which we call the locking method, dynamically keeps track of grid points that have either already had the solution successfully calculated at that grid point or for which the solution cannot be successfully calculated during the current iteration. These locked points can quickly be skipped over during the fast sweeping iterations, avoiding many time-consuming calculations. The second improvement, which we call the two queue method, keeps all of the unlocked points in a data structure so that the locked points no longer need to be visited at all. Unfortunately, it is not possible to insert new points into the data structure while maintaining the fast sweeping ordering without at least occasionally sorting. Instead, we segregate the grid points into those with small predicted solutions and those with large predicted solutions using two queues. We give two ways of performing this segregation. This method is a label correcting (iterative) method like the fast sweeping method, but it tends to operate near the front like the fast marching method. It is reminiscent of the threshold method for finding the shortest path on a network, [F. Glover, D. Klingman, and N. Phillips, Oper. Res., 33 (1985), pp. 65-73]. We demonstrate the numerical efficiency of the improved methods on a number of examples.

An Ordered Upwind Method with Precomputed Stencil and Monotone Node Acceptance for Solving Static Convex Hamilton-Jacobi Equations

“…The computed solution for levels 13 and 18 grids are shown in Figure 5. 4. Experiments indicate that refining the grid along the sinusoidal layer boundaries does not improve accuracy significantly.…”

“…There are also a number of sweeping algorithms which use dynamic node orderings; for example [20,6,4]. These algorithms attempt to approximate the optimal ordering generated by single-pass methods such as FMM without the overhead associated with managing an accurate queue.…”

“…We define the numerical Hamiltonian as follows: 4) where ∈ Ω, is a set of simplices in Ω ∪ ∂Ω such that / ∈ and vectors − are independent, : Ω → ℝ is bounded, and ∈ ℝ. When we are only varying , we write ( ) = ( , , , ) for convenience.…”

Joint transmission and reflection traveltime tomography using the fast sweeping method and the adjoint-state technique

“…2003; Zhao 2004). The implementation of FSM is thus significantly simplified and its computational efficiency is superior to FMM even when velocity changes rapidly (Bak et al . 2010).…”

“…FSM advances wave fronts with iterative sweeping and updates traveltime at each grid point monotonically to ensure the causality (Tsai et al 2003;Zhao 2004). The implementation of FSM is thus significantly simplified and its computational efficiency is superior to FMM even when velocity changes rapidly (Bak et al 2010). In addition, FSM can be easily extended to high-order accuracy (Zhang et al 2006;Xiong et al 2010), achieve high computational performance on shared or distributed memory systems (Zhao 2007) and can be implemented on unstructured elements as well (Qian et al 2007).…”