High-Order Factorization Based High-Order Hybrid Fast Sweeping Methods for Point-Source Eikonal Equations

Abstract

The solution for the eikonal equation with a point-source condition has an upwind singularity at the source point as the eikonal solution behaves like a distance function at and near the source. As such, the eikonal function is not differentiable at the source so that all formally high-order numerical schemes for the eikonal equation yield first-order convergence and relatively large errors. Therefore, it is a longstanding challenge in computational geometrical optics how to compute a uniformly high-order accurate solution for the point-source eikonal equation in a global domain. In this paper, assuming that both the squared slowness and the squared eikonal are analytic near the source, we propose high-order factorization based high-order hybrid fast sweeping methods for point-source eikonal equations to compute just such solutions. Observing that the squared eikonal is differentiable at the source, we propose to factorize the eikonal into two multiplicative or additive factors, one of which is specified to approximate the eikonal up to arbitrary order of accuracy near the source, and the other of which serves as a higher-order correction term. This decomposition is achieved by using the eikonal equation and applying power series expansions to both the squared eikonal and the squared slowness function. We develop recursive formulas to compute the approximate eikonal up to arbitrary order of accuracy near the source. Furthermore, these approximations enable us to decompose the eikonal into two factors, either multiplicatively or additively, so that we can design two new types of hybrid, high-order fast sweeping schemes for the point-source eikonal equation. We also show that the first-order hybrid fast sweeping methods are monotone and consistent so that they are convergent in computing viscosity solutions. Two-and three-dimensional numerical examples demonstrate that a hybrid pth order fast sweeping method yields desired, uniform, clean pth order convergence in a global domain by using a pth order factorization.

A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

“…In practice, such difficulties in initializing amplitudes are handled in an ad hoc manner. The amplitudes are initialized a little bit away from the point source using amplitudes for a medium with constant refractive index [71,60,47,49,48]; consequently, the resulting numerical asymptotic solution is not uniform near the source.…”

Convergence analysis of the fast sweeping method for static convex Hamilton–Jacobi equations

“…The fast sweeping method is an efficient iterative method for solving (1.1). It has been developed and used successfully for various hyperbolic problems (e.g., see [6,7,14,15,17,20,23,26,28,30,34,35,40,[43][44][45] and references therein for development, and [3,18,21,24,25,27,29,31] and references therein for applications to different problems). The key ingredients for the success of the FSM are the following:…”

Binary recovery via phase field regularization for first-arrival traveltime tomography

“…Numerical methods have been investigated to obtain unique viscosity solutions, typically using monotone finite difference schemes for stability. We use the Fast Marching Method (FMM) [47,48,46,1] due to its robustness [28] though fast sweeping methods [53,33,42,30] are also popular. arXiv:1811.02865v2 [math.NA] 9 Nov 2018…”