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

  • M. Baumann
  • R. Astudillo
  • Y. Qiu
  • E. Y. M. Ang
  • M. B. van Gijzen
  • R.-É. Plessix
Open Access
Original Paper

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.

Keywords

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

Notes

Acknowledgments

We would like to thank Joost van Zwieten, co-developer of the open source project nutils1 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.

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)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle 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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  10. 10.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis. Towards integration of CAD and FEA. John Wiley & Son Ltd. (2009)Google Scholar
  11. 11.
    De Basabe, J.: High-order Finite Element Methods for Seismic Wave Propagation. Ph.D. thesis The University of Texas at Austin (2009)Google Scholar
  12. 12.
    Dewilde, P., Van der Veen, A.: Time-Varying Systems and Computations. Kluwer Academic Publishers, Boston (1998)Google Scholar
  13. 13.
    Eidelman, Y., Gohberg, I.: On generators of quasiseparable finite block matrices. Calcolo 42(3), 187–214 (2005)CrossRefGoogle 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)Google Scholar
  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)CrossRefGoogle 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)Google Scholar
  17. 17.
    Jbilou, K., Messaoudi, A., Sadok, H.: Global FOM and GMRES algorithms for matrix equations. Appl. Numer. Math. 31, 49–63 (1999)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle Scholar
  20. 20.
    Liesen, J., Strakos, Z.: Krylov subspace methods: Principles and analysis. Numerical mathematics and scientific computation OUP Oxford (2013)Google Scholar
  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)CrossRefGoogle 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)CrossRefGoogle Scholar
  24. 24.
    Plessix, R.E.: A Helmholtz iterative solver for 3D seismic-imaging problems. Geophysics 72(5), SM185–SM194 (2007)CrossRefGoogle Scholar
  25. 25.
    Plessix, R.E.: Three-dimensional frequency-domain full-waveform inversion with an iterative solver. Geophysics 74, WCC149–WCC157 (2009)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle 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)Google Scholar
  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)Google Scholar
  31. 31.
    Rice, J.: Efficient Algorithms for Distributed Control: a Structured Matrix Approach. Ph.D. thesis, Delft University of Technology (2010)Google Scholar
  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)CrossRefGoogle 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)CrossRefGoogle 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)CrossRefGoogle 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)Google Scholar
  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)CrossRefGoogle 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)CrossRefGoogle 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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  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)CrossRefGoogle Scholar
  43. 43.
    Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices: Linear Systems. Johns Hopkins University Press, Baltimore (2007)Google Scholar
  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)Google Scholar
  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)Google Scholar
  47. 47.
    Xia, J.: Efficient structured multifrontal factorization for general large sparse matrices. SIAM J. Sci. Comput. 35(2), A832–A860 (2013)CrossRefGoogle Scholar

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© The Author(s) 2017

Open AccessThis 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.

Authors and Affiliations

  1. 1.Delft University of TechnologyDelftThe Netherlands
  2. 2.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  3. 3.Nanyang Technological UniversitySingaporeSingapore
  4. 4.Shell Global Solutions International B.V.The HagueThe Netherlands

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