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Sweeping preconditioners for elastic wave propagation withspectral element methods

Published online by Cambridge University Press:  20 February 2014

Paul Tsuji ,
Jack Poulson ,
Björn Engquist  and
Lexing Ying
Paul Tsuji
Affiliation:
Sandia National Laboratories, Org. 1442: Numerical Analysis and Applications, Livermore, CA 94550, USA.
Jack Poulson
Affiliation:
Georgia Institute of Technology, School of Computational Science and Engineering, Atlanta, GA 30332, USA.
Björn Engquist
Affiliation:
University of Texas at Austin, Department of Mathematics, Austin, TX 78712, USA.
Lexing Ying
Affiliation:
Stanford University, Department of Mathematics, Stanford, CA 94305, USA.. [email protected]
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Abstract

We present a parallel preconditioning method for the iterative solution of thetime-harmonic elastic wave equation which makes use of higher-order spectral elements toreduce pollution error. In particular, the method leverages perfectly matched layerboundary conditions to efficiently approximate the Schur complement matrices of a blockLDLTfactorization. Both sequential and parallel versions of the algorithm are discussed andresults for large-scale problems from exploration geophysics are presented.

Keywords

Elastic waveseismic wavetime-harmonicfrequency domainspectral elementsparallel preconditioneriterative solversparse-directperfectly matched layersfull waveform inversion
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 433 - 447
Copyright
© EDP Sciences, SMAI, 2014

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References

F. Aminzadeh, J. Brac and T. Kunz, 3-D Salt and Overthrust models. In SEG/EAGE 3-D Modeling Series 1. Tulsa, OK (1997). Google Scholar
Berenger, J.P., A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185–200. Google Scholar
Bramble, J. and Pasciak, J., A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: a priori estimates in H 1. J. Math Anal. Appl. 345 (2008) 396–404. Google Scholar
Chew, W.C. and Weedon, W.H., A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microwave Optical Tech. Lett. 7 (1994) 599–604. Google Scholar
J. Choi, J.J. Dongarra, R. Pozo and D.W. Walker, ScaLAPACK: A scalable linear algebra library for distributed memory concurrent computers, in Proc. of the Fourth Symposium on the Frontiers of Massively Parallel Computation, IEEE Comput. Soc. Press(1992) 120–127. Google Scholar
Engquist, B. and Ying, L., Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation. Commun. Pure Appl. Math. 64 (2011) 697–735. Google Scholar
Engquist, B. and Ying, L., Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul. 9 (2011) 686–710. Google Scholar
Erlangga, Y.A., Vuik, C. and Oosterlee, C.W., On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math. 50 (2004) 409–425. Google Scholar
O.G. Ernst and M.J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, in vol. 83 of Numerical Analysis of Multiscale Problems. Edited by I. Graham, T. Hou, O. Lakkis and R. Scheichl. Springer-Verlag (2011) 325–361. CrossRefGoogle Scholar
Grasedyck, L. and Hackbusch, W., Construction and arithmetics of ℋ-matrices. Computing 70 (2003) 295–334. Google Scholar
Gupta, A., Karypis, G. and Kumar, V., A highly scalable parallel algorithm for sparse matrix factorization. IEEE Transactions on Parallel and Distributed Systems 8 (1997) 502–520. Google Scholar
A. Gupta, S. Koric and T. George, Sparse matrix factorization on massively parallel computers, in Proc. of the Conference on High Performance Computing, Networking, Storage and Analysis. Portland, OR (2009). CrossRefGoogle Scholar
Harari, I. and Albocher, U., Studies of FE/PML for exterior problems of time-harmonic elastic waves. Comput. Methods Appl. Mech. Eng. 195 (2006) 3854–3879. Google Scholar
T. Hughes, The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Inc. (1987). Google Scholar
G. Karniadakis, Spectral/hp element methods for CFD. Oxford University Press (1999). Google Scholar
Komatitsch, D. and Tromp, J., Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int. 139 (1999) 806–822. Google Scholar
Liu, J.. The multifrontal method for sparse matrix solution: theory and practice. SIAM Review 34 (1992) 82–109. Google Scholar
Patera, A., A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54 (1984) 468–488. Google Scholar
J. Poulson, B. Engquist, S. Li and L. Ying, A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations e-prints ArXiv (2012). CrossRefGoogle Scholar
Poulson, J., Marker, B., van de Geijn, R.A., Hammond, J.R. and Romero, N.A., Elemental: A new framework for distributed memory dense matrix computations. ACM Trans. Math. Software 39. Google Scholar
Raghavan, P., Efficient parallel sparse triangular solution with selective inversion. Parallel Proc. Lett. 8 (1998) 29–40. Google Scholar
Saad, Y. and Schultz, M.H., A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7 (1986) 856–869. Google Scholar
Schreiber, R., A new implementation of sparse Gaussian elimination. ACM Trans. Math. Software 8 (1982) 256–276. Google Scholar
Tsuji, P., Engquist, B. and Ying, L., A sweeping preconditioner for time-harmonic Maxwell’s equations with finite elements. J. Comput. Phys. 231 (2012) 3770–3783. Google Scholar
Tsuji, P. and Ying, L., A sweeping preconditioner for Yee’s finite difference approximation of time-harmonic Maxwell’s equations. Frontiers of Mathematics in China 7 (2012) 347–363. Google Scholar