Geodesic
A 3D Helmholtz iterative solver with a semi-analytical preconditioner for acoustic wavefield modeling in seismic exploration problems
Numerical methods and programming, Tome 15 (2014) no. 3, pp. 514-529.

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An approach to the iterative solution of the 3D acoustic wave equation in a frequency domain is proposed, substantiated, and verified numerically. Our method is based on Krylov-type linear solvers, similarly to several other iterative solver approaches. The distinctive feature of our method is the use of a right preconditioner obtained as the solution of the complex dumped Helmholtz equation in a 1D medium, where velocities vary only with depth. The actual Helmholtz operator is represented as a perturbation of the preconditioner. As a result, a matrix-by-vector multiplication of the preconditioned system can be efficiently evaluated via 2D FFT in x and y directions followed by the solution of a number of ordinary differential equations in z directions. While solving ODE's. it is possible to treat the 1D velocity function as a piecewise constant one and to search for the exact solution as a superposition of upgoing and downgoing waves. This approach allows one not to use explicit finite-difference approximations of derivatives at all. The method has excellent dispersion properties in both lateral and vertical directions.
Keywords: Helmholtz equation, iterative methods, preconditioners, acoustic waves, seismic exploration. 
@article{VMP_2014_15_3_a11,
     author = {D. A. Neklyudov and I. Yu. Silvestrov and V. A. Tcheverda},
     title = {A {3D} {Helmholtz} iterative solver with a semi-analytical preconditioner for acoustic wavefield modeling in seismic exploration problems},
     journal = {Numerical methods and programming},
     pages = {514--529},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2014_15_3_a11/}
}
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D. A. Neklyudov; I. Yu. Silvestrov; V. A. Tcheverda. A 3D Helmholtz iterative solver with a semi-analytical preconditioner for acoustic wavefield modeling in seismic exploration problems. Numerical methods and programming, Tome 15 (2014) no. 3, pp. 514-529. http://geodesic.mathdoc.fr/item/VMP_2014_15_3_a11/