# Sweeping preconditioners for elastic wave propagation with spectral element methods

Paul Tsuji; Jack Poulson; Björn Engquist; Lexing Ying

- Volume: 48, Issue: 2, page 433-447
- ISSN: 0764-583X

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topTsuji, Paul, et al. "Sweeping preconditioners for elastic wave propagation with spectral element methods." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.2 (2014): 433-447. <http://eudml.org/doc/273345>.

@article{Tsuji2014,

abstract = {We present a parallel preconditioning method for the iterative solution of the time-harmonic elastic wave equation which makes use of higher-order spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LDLT factorization. Both sequential and parallel versions of the algorithm are discussed and results for large-scale problems from exploration geophysics are presented.},

author = {Tsuji, Paul, Poulson, Jack, Engquist, Björn, Ying, Lexing},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {elastic wave; seismic wave; time-harmonic; frequency domain; spectral elements; parallel preconditioner; iterative solver; sparse-direct; perfectly matched layers; full waveform inversion},

language = {eng},

number = {2},

pages = {433-447},

publisher = {EDP-Sciences},

title = {Sweeping preconditioners for elastic wave propagation with spectral element methods},

url = {http://eudml.org/doc/273345},

volume = {48},

year = {2014},

}

TY - JOUR

AU - Tsuji, Paul

AU - Poulson, Jack

AU - Engquist, Björn

AU - Ying, Lexing

TI - Sweeping preconditioners for elastic wave propagation with spectral element methods

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 2

SP - 433

EP - 447

AB - We present a parallel preconditioning method for the iterative solution of the time-harmonic elastic wave equation which makes use of higher-order spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LDLT factorization. Both sequential and parallel versions of the algorithm are discussed and results for large-scale problems from exploration geophysics are presented.

LA - eng

KW - elastic wave; seismic wave; time-harmonic; frequency domain; spectral elements; parallel preconditioner; iterative solver; sparse-direct; perfectly matched layers; full waveform inversion

UR - http://eudml.org/doc/273345

ER -

## References

top- [1] F. Aminzadeh, J. Brac and T. Kunz, 3-D Salt and Overthrust models. In SEG/EAGE 3-D Modeling Series 1. Tulsa, OK (1997).
- [2] J.P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys.114 (1994) 185–200. Zbl0814.65129MR1294924
- [3] J. Bramble and J. Pasciak, A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: a priori estimates in H1. J. Math Anal. Appl.345 (2008) 396–404. Zbl1146.35323MR2422659
- [4] W.C. Chew and W.H. Weedon, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microwave Optical Tech. Lett.7 (1994) 599–604.
- [5] 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. Zbl0926.65148
- [6] B. Engquist and L. Ying, Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation. Commun. Pure Appl. Math.64 (2011) 697–735. Zbl1229.35037MR2789492
- [7] B. Engquist and L. Ying, Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul.9 (2011) 686–710. Zbl1228.65234MR2818416
- [8] Y.A. Erlangga, C. Vuik and C.W. Oosterlee, On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math.50 (2004) 409–425. Zbl1051.65101MR2074012
- [9] 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. Zbl1248.65128MR3050918
- [10] L. Grasedyck and W. Hackbusch, Construction and arithmetics of ℋ-matrices. Computing 70 (2003) 295–334. Zbl1030.65033MR2011419
- [11] A. Gupta, G. Karypis and V. Kumar, A highly scalable parallel algorithm for sparse matrix factorization. IEEE Transactions on Parallel and Distributed Systems8 (1997) 502–520.
- [12] 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).
- [13] I. Harari and U. Albocher, Studies of FE/PML for exterior problems of time-harmonic elastic waves. Comput. Methods Appl. Mech. Eng.195 (2006) 3854–3879. Zbl1119.74048MR2221777
- [14] T. Hughes, The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Inc. (1987). Zbl0634.73056MR1008473
- [15] G. Karniadakis, Spectral/hp element methods for CFD. Oxford University Press (1999). Zbl0954.76001MR1696933
- [16] D. Komatitsch and J. Tromp, Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int.139 (1999) 806–822.
- [17] J. Liu. The multifrontal method for sparse matrix solution: theory and practice. SIAM Review34 (1992) 82–109. Zbl0919.65019MR1156290
- [18] A. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys.54 (1984) 468–488. Zbl0535.76035
- [19] J. Poulson, B. Engquist, S. Li and L. Ying, A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations e-prints ArXiv (2012). Zbl1275.65073MR3048222
- [20] J. Poulson, B. Marker, R.A. van de Geijn, J.R. Hammond and N.A. Romero, Elemental: A new framework for distributed memory dense matrix computations. ACM Trans. Math. Software 39. Zbl1295.65137MR3031632
- [21] P. Raghavan, Efficient parallel sparse triangular solution with selective inversion. Parallel Proc. Lett.8 (1998) 29–40. MR1632870
- [22] Y. Saad and M.H. Schultz, A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput.7 (1986) 856–869. Zbl0599.65018MR848568
- [23] R. Schreiber, A new implementation of sparse Gaussian elimination. ACM Trans. Math. Software8 (1982) 256–276. Zbl0491.65013MR695356
- [24] P. Tsuji, B. Engquist and L. Ying, A sweeping preconditioner for time-harmonic Maxwell’s equations with finite elements. J. Comput. Phys.231 (2012) 3770–3783. Zbl1251.78013MR2902419
- [25] P. Tsuji and L. Ying, A sweeping preconditioner for Yee’s finite difference approximation of time-harmonic Maxwell’s equations. Frontiers of Mathematics in China7 (2012) 347–363. Zbl1253.78049MR2897708

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