Separation-of-variables as a preconditioner for an iterative Helmholtz solver. (English) Zbl 1013.65117

Summary: A preconditioned iterative method based on separation-of-variables for solving the Helmholtz equation in an inhomogeneous medium is tested. The preconditioner is constructed by approximating the wavenumber by a sum of two terms, one depending only on one spatial coordinate, say \(x\), and the other depending on the remaining coordinates. The Helmholtz equation can be solved efficiently if the wavenumber has such a separable form.
First, an eigenvalue-eigenvector decomposition is applied in the \(x\)-direction. With these eigenvectors, a change of variables is performed in order to obtain a set of independent systems with one dimension less than the original one. For smooth models and low frequencies, the convergence rate with this preconditioner is satisfactory. Unfortunately, it rapidly deteriorates when the roughness of the model or the frequency increases.
Examples from seismic modeling are given to illustrate this behaviour. Moreover, numerical evidence is presented that suggests that the decomposition of the wavenumber in the sum of two terms cannot be improved with this approach.


65N06 Finite difference methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
86A15 Seismology (including tsunami modeling), earthquakes


Full Text: DOI


[1] Bamberger, A; Joly, P; Roberts, J.E, Second-order absorbing boundary conditions for the wave equation: A solution for the corner problem, SIAM J. numer. anal., 27, 323-352, (1990) · Zbl 0716.35036
[2] Bourgeois, A; Bourget, M; Lailly, P; Poulet, M; Ricarte, P; Versteeg, R, Marmousi, model and data, (), 5-16
[3] Bramble, J.H; Pasiack, J.E; Xu, J, The analysis of multigrid algorithms for nonsymmetric and indefinite problems, Math. comp., 51, 389, (1988)
[4] Elman, H.C; O’Leary, D.P, Efficient iterative solution of the three-dimensional Helmholtz equation, J. comput. phys., 142, 163-181, (1998) · Zbl 0929.65089
[5] George, A; Liu, J.W, Computer solution of large sparse positive definite systems, (1981), Prentice-Hall Englewood Cliffs, NJ · Zbl 0516.65010
[6] Heikkola, E; Kuznetsov, Y.A; Lipnikov, K.N, Fictitious domain methods for the numerical solution of three-dimensional acoustic scattering problems, J. comput. acoust., 7, 3, 161-183, (1999) · Zbl 1360.76140
[7] Larson, E, A domain decomposition method for the Helmholtz equation in a multilayer domain, SIAM J. sci. comput., 20, 5, 1713-1731, (1999) · Zbl 0936.65140
[8] Reusken, A, On the approximate cyclic reduction preconditioner, SIAM J. sci. comput., 21, 2, 565-590, (1999) · Zbl 0943.65046
[9] Rossi, T; Toivanen, J, A parallel fast direct solver for block tridiagonal systems with separable matrix of arbitrary dimension, SIAM J. sci. comput., 1778-1796, (1999) · Zbl 0931.65020
[10] Saad, Y; Schultz, M.H, GMRES: A generalized minimal residual method for solving nonsymmetric linear system, SIAM J. sci. statist. comput., 7, 856-869, (1986) · Zbl 0599.65018
[11] van der Vorst, H.A, {\scbi-CGSTAB}: A fast and smoothly converging variant of bi-CG for the solution of nonsymmetric linear systems, SIAM J. sci. statist. comput., 13, 631-644, (1992) · Zbl 0761.65023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.