ForwardHelmholtz.jl
swMATH ID:  46001 
Software Authors:  Treister, Eran; Haber, Eldad 
Description:  Julia package ForwardHelmholtz.jl: A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude. The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complexvalued advectiondiffusionreaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our secondorder advectiondiffusionreaction discretization is more accurate than the standard secondorder discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach. 
Homepage:  https://arxiv.org/abs/1712.06091 
Source Code:  https://github.com/JuliaInv/ForwardHelmholtz.jl 
Dependencies:  Julia 
Keywords:  factored eikonal equation; fast marching; Helmholtz equation; multigrid; seismic modeling; shifted Laplacian 
Related Software:  Julia; clique; PSP; CIFAR; ImageNet; UNet; DeepXDE; Adam; jInv 
Cited in:  3 Documents 
Standard Articles
1 Publication describing the Software, including 1 Publication in zbMATH  Year 

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude. Zbl 1524.65929 Treister, Eran; Haber, Eldad 
2019

Cited by 5 Authors
2  Treister, Eran 
1  Azulay, Yael 
1  Haber, Eldad 
1  Huang, Guangnan 
1  Luo, Songting 
Cited in 3 Serials
1  Journal of Scientific Computing 
1  SIAM Journal on Scientific Computing 
1  Numerical Linear Algebra with Applications 
all
top 5
Cited in 6 Fields
3  Numerical analysis (65XX) 
2  Geophysics (86XX) 
1  Partial differential equations (35XX) 
1  Computer science (68XX) 
1  Mechanics of deformable solids (74XX) 
1  Fluid mechanics (76XX) 