Skip to main content

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

Abstract

In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.

Notes

  1. 1.

    http://www.nutils.org/

References

  1. 1.

    Airaksinen, T., Pennanen, A., Toivanen, J.: A damping preconditioner for time-harmonic wave equations in fluid and elastic material. J. Comput. Phys. 228(5), 1466–1479 (2009)

    Article  Google Scholar 

  2. 2.

    Amestoy, P., Ashcraft, C., Boiteau, O., Buttari, A., L’Excellent, J.Y., Weisbecker, C.: Improving multifrontal methods by means of block low-rank representations. SIAM J. Sci. Comput. 37, A1451–A1474 (2015)

    Article  Google Scholar 

  3. 3.

    Amestoy, P., Brossier, R., Buttari, A., L’Excellent, J.Y., Mary, T., Métivier, L., Miniussi, A., Operto, S.: Fast 3D frequency-domain full-waveform inversion with a parallel block low-rank multifrontal direct solver: Application to OBC data from the North Sea. Geophysics 81(6), R363–R383 (2016)

    Article  Google Scholar 

  4. 4.

    Astudillo, R., van Gijzen, M.B.: Induced dimension reduction method for solving linear matrix equations. Procedia Computer Science 80, 222–232 (2016)

  5. 5.

    Baumann, M.: Two benchmark problems for the time-harmonic elastic wave equation in 2D and 3D. doi:http://dx.doi.org/https://github.com/ManuelMBaumann/elastic_benchmarks (Sept. 2016). doi:10.5281/zenodo.154700

  6. 6.

    Baumann, M., van Gijzen, M.B.: Nested Krylov methods for shifted linear systems. SIAM J. Sci. Comput. 37(5), S90–S112 (2015)

  7. 7.

    Baumann, M., van Gijzen, M.B.: An efficient two-level preconditioner for multi-frequency wave propagation problems. Tech. rep., DIAM Report 17-03 Delft University of Technology (2017)

  8. 8.

    Chandrasekaran, S., Dewilde, P., Gu, M., Pals, T., Sun, X., van der Veen, A., White, D.: Some fast algorithms for sequentially semiseparable representations. SIAM J. Matrix Anal. Appl. 27(2), 341–364 (2005)

  9. 9.

    Chandrasekaran, S., Dewilde, P., Gu, M., Pals, T., van der Veen, A.J.: Fast Stable Solvers for Sequentially Semi-separable Linear Systems of Equations. Tech. rep., Lawrence Livermore National Laboratory (2003)

  10. 10.

    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis. Towards integration of CAD and FEA. John Wiley & Son Ltd. (2009)

  11. 11.

    De Basabe, J.: High-order Finite Element Methods for Seismic Wave Propagation. Ph.D. thesis The University of Texas at Austin (2009)

  12. 12.

    Dewilde, P., Van der Veen, A.: Time-Varying Systems and Computations. Kluwer Academic Publishers, Boston (1998)

  13. 13.

    Eidelman, Y., Gohberg, I.: On generators of quasiseparable finite block matrices. Calcolo 42(3), 187–214 (2005)

    Article  Google Scholar 

  14. 14.

    Elman, H., Silvester, D., Wathen, A.: Finite elements and fast iterative solvers: With applications in incompressible fluid dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press (2014)

  15. 15.

    Etienne, V., Chaljub, E., Virieux, J., Glinsky, N.: An hp-adaptive discontinuous Galerkin finite-element method for 3-D elastic wave modelling. Geophys. J. Int. 183(2), 941–962 (2010)

    Article  Google Scholar 

  16. 16.

    van Gijzen, M.B., Sonneveld, P.: Algorithm 913: An Elegant IDR(s) Variant that Efficiently Exploits Bi-orthogonality Properties. ACM Trans. Math. Software 38(1), 5:1–5:19 (2011)

  17. 17.

    Jbilou, K., Messaoudi, A., Sadok, H.: Global FOM and GMRES algorithms for matrix equations. Appl. Numer. Math. 31, 49–63 (1999)

    Article  Google Scholar 

  18. 18.

    Kavcic, A., Moura, J.: Matrices with banded inverses: inversion algorithms and factorization of Gauss-Markov processes. IEEE Trans. Inform. Theory 46(4), 1495–1509 (2000)

    Article  Google Scholar 

  19. 19.

    Knibbe, H., Vuik, C., Oosterlee, C.W.: Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units. Comput. Geosci. 20(2), 297–315 (2016)

    Article  Google Scholar 

  20. 20.

    Liesen, J., Strakos, Z.: Krylov subspace methods: Principles and analysis. Numerical mathematics and scientific computation OUP Oxford (2013)

  21. 21.

    Martin, G.S., Marfurt, K.J., Larsen, S.: Marmousi-2: an Updated Model for the Investigation of AVO in Structurally Complex Areas 72Nd Annual International Meeting, SEG, Expanded Abstract, pp 1979–1982 (2002)

    Google Scholar 

  22. 22.

    Mulder, W.A., Plessix, R.E.: How to choose a subset of frequencies in frequency-domain finite-difference migration. Geophys. J. Int. 158, 801–812 (2004)

    Article  Google Scholar 

  23. 23.

    Petrov, P.V., Newman, G.A.: Three-dimensional inverse modelling of damped elastic wave propagation in the Fourier domain. Geophys. J. Int. 198, 1599–1617 (2014)

    Article  Google Scholar 

  24. 24.

    Plessix, R.E.: A Helmholtz iterative solver for 3D seismic-imaging problems. Geophysics 72(5), SM185–SM194 (2007)

    Article  Google Scholar 

  25. 25.

    Plessix, R.E.: Three-dimensional frequency-domain full-waveform inversion with an iterative solver. Geophysics 74, WCC149–WCC157 (2009)

    Article  Google Scholar 

  26. 26.

    Plessix, R.E., Mulder, W.A.: Seperation-of-variables as a preconditioner for an iterative Helmholtz solver. Appl. Numer. Math. 44, 385–400 (2004)

    Article  Google Scholar 

  27. 27.

    Plessix, R.E., Pérez Solano, C. A.: Modified surface boundary conditions for elastic waveform inversion of low-frequency wide-angle active land seismic data. Geophys. J. Int. 201, 1324–1334 (2015)

    Article  Google Scholar 

  28. 28.

    Pratt, R.: Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale mode. Geophysics 64(3), 888–901 (1999)

    Article  Google Scholar 

  29. 29.

    Qiu, Y., van Gijzen, M.B., van Wingerden, J.W., Verhaegen, M., Vuik, C.: Efficient Preconditioners for PDE-constrained Optimization Problems with a Multilevel Sequentially SemiSeparable Matrix Structure. Electron. Trans. Numer. Anal. 44, 367–400 (2015)

  30. 30.

    Qiu, Y., van Gijzen, M.B., van Wingerden, J.W., Verhaegen, M., Vuik, C.: Evaluation of multilevel sequentially semiseparable preconditioners on CFD benchmark problems using incompressible flow and iterative solver software. Math. Methods Appl. Sci. 38 (2015)

  31. 31.

    Rice, J.: Efficient Algorithms for Distributed Control: a Structured Matrix Approach. Ph.D. thesis, Delft University of Technology (2010)

  32. 32.

    Rice, J., Verhaegen, M.: Distributed control: a sequentially Semi-Separable approach for spatially heterogeneous linear systems. IEEE Trans. Automat. Control 54(6), 1270–1283 (2009)

    Article  Google Scholar 

  33. 33.

    Riyanti, C.D., Erlangga, Y.A., Plessix, R.E., Mulder, W.A., Vuik, C., Osterlee, C.: A new iterative solver for the time-harmonic wave equation. Geophysics 71, E57–E63 (2006)

    Article  Google Scholar 

  34. 34.

    Rizzuti, G., Mulder, W.: Multigrid-based ’shifted-Laplacian’ preconditioning for the time-harmonic elastic wave equation. J. Comput. Phys. 317, 47–65 (2016)

    Article  Google Scholar 

  35. 35.

    Saad, Y.: SPARSEKIT: a Basic Tool Kit for Sparse Matrix Computations. Tech. Rep. University of Minnesota, Minneapolis (1994)

    Google Scholar 

  36. 36.

    Saad, Y.: Iterative methods for sparse linear systems: Second edition. Society for Industrial and Applied Mathematics (2003)

  37. 37.

    Saad, Y., Schultz, M.: GMRES: A generalized minimal residual algorithm for solving nonsymetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  Google Scholar 

  38. 38.

    Saibaba, A., Bakhos, T., Kitanidis, P.: A flexible Krylov solver for shifted systems with application to oscillatory hydraulic tomography. SIAM J. Sci. Comput. 35, 3001–3023 (2013)

    Article  Google Scholar 

  39. 39.

    Sleijpen, G.L.G., Sonneveld, P., van Gijzen, M.B.: BiCGStab as an induced dimension reduction method. Appl. Numer. Math. 60, 1100–1114 (2010)

  40. 40.

    Sleijpen, G.L.G., van der Vorst, H.A.: Maintaining convergence properties of BiCGstab methods in finite precision arithmetic. Numer. Algorithms 10, 203–223 (1995)

  41. 41.

    Sonneveld, P., van Gijzen, M.B.: IDR(S): a family of simple and fast algorithms for solving large nonsymmetric linear systems. SIAM J. Sci. Comput. 31(2), 1035–1062 (2008)

  42. 42.

    Tsuji, P., Poulson, J., Engquist, B., Ying, L.: Sweeping preconditioners for elastic wave propagation with spectral element methods. ESAIM Math. Model. Numer. Anal. 48(2), 433–447 (2014)

    Article  Google Scholar 

  43. 43.

    Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices: Linear Systems. Johns Hopkins University Press, Baltimore (2007)

  44. 44.

    Virieux, J., Operto, S.: An overview of full-waveform inversion in exploration geophysics. Geophysics 73 (6), VE135–VE144 (2009)

    Google Scholar 

  45. 45.

    van der Vorst, H.A.: BiCGStab: A Fast and Smoothly Converging Variant of bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)

  46. 46.

    Wang, S., de Hoop, M.V., Xia, J., Li, X.: Massively parallel structured multifrontal solver for time-harmonic elastic waves in 3-D anisotropic media. Geophys. J. Int. 191(1), 346–366 (2012)

  47. 47.

    Xia, J.: Efficient structured multifrontal factorization for general large sparse matrices. SIAM J. Sci. Comput. 35(2), A832–A860 (2013)

    Article  Google Scholar 

Download references

Acknowledgments

We would like to thank Joost van Zwieten, co-developer of the open source project nutils Footnote 1 for helpful discussions concerning the finite element discretization described in Section 2.2. Shell Global Solutions International B.V. is gratefully acknowledged for financial support of the first author.

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. Baumann.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Baumann, M., Astudillo, R., Qiu, Y. et al. An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies. Comput Geosci 22, 43–61 (2018). https://doi.org/10.1007/s10596-017-9667-7

Download citation

Keywords

  • Time-harmonic elastic wave equation
  • Multiple frequencies
  • Induced dimension reduction (IDR) method
  • Preconditioned matrix equations
  • Multilevel sequentially semiseparable (MSSS) matrices