Geodesic
Two-level preconditioned Krylov subspace methods for the solution of three-dimensional heterogeneous Helmholtz problems in seismics
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 2, pp. 213-221

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In this paper we address the solution of three-dimensional heterogeneous Helmholtz problems discretized with compact fourth-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 linear systems of equations. We propose an iterative two-grid method where the coarse grid problem is solved inexactly. A single cycle of this method is used as a variable preconditioner for a flexible Krylov subspace method. Numerical results demonstrate the usefulness of the algorithm on a realistic three-dimensional application. The proposed numerical method allows us to solve wave propagation problems with single or multiple sources even at high frequencies on a reasonable number of cores of a distributed memory cluster. 
@article{SJVM_2012_15_2_a11,
     author = {H. Calandra and S. Gratton and R. Lago and X. Pinel and X. Vasseur},
     title = {Two-level preconditioned {Krylov} subspace methods for the solution of three-dimensional heterogeneous {Helmholtz} problems in seismics},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {213--221},
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     volume = {15},
     number = {2},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2012_15_2_a11/}
}
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H. Calandra; S. Gratton; R. Lago; X. Pinel; X. Vasseur. Two-level preconditioned Krylov subspace methods for the solution of three-dimensional heterogeneous Helmholtz problems in seismics. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 2, pp. 213-221. http://geodesic.mathdoc.fr/item/SJVM_2012_15_2_a11/