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An improved two-grid preconditioner for the solution of three-dimensional Helmholtz problems in heterogeneous media


  • Paper submitted for the special issue of Numerical Linear Algebra with Applications related to the OPTPDE ESFWaves workshop held in Würzburg, Germany, on 26–28 September 2011. Revised version.

Correspondence to: Xavier Vasseur, CERFACS and HiePACS project joint INRIA-CERFACS Laboratory, 42 Avenue Gaspard Coriolis, F-31057 Toulouse Cedex 1, France.



In this paper, we address the solution of three-dimensional heterogeneous Helmholtz problems discretized with second-order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large indefinite linear systems of equations. For that purpose, we propose and analyse an iterative two-grid method acting on the Helmholtz operator where the coarse grid problem is solved inaccurately. A cycle of a multigrid method applied to a complex shifted Laplacian operator is used as a preconditioner for the approximate solution of this coarse problem. A single cycle of the new method is then used as a variable preconditioner of a flexible Krylov subspace method. We analyse the properties of the resulting preconditioned operator by Fourier analysis. Numerical results demonstrate the effectiveness of the algorithm on three-dimensional applications. The proposed numerical method allows us to solve three-dimensional wave propagation problems even at high frequencies on a reasonable number of cores of a distributed memory computer. Copyright © 2012 John Wiley & Sons, Ltd.