A Geometric Multigrid Preconditioner for the Solution of the Helmholtz Equation in Three-Dimensional Heterogeneous Media on Massively Parallel Computers

Chapter
First Online:
Part of the Geosystems Mathematics book series (GSMA)

Abstract

We consider the numerical simulation of acoustic wave propagation in three-dimensional heterogeneous media as occurring in seismic exploration. We focus on forward Helmholtz problems written in the frequency domain, since this setting is known to be particularly challenging for modern iterative methods. The geometric multigrid preconditioner proposed by Calandra et al. (Numer Linear Algebra Appl 20:663–688, 2013) is considered for the approximate solution of the Helmholtz equation at high frequencies in combination with dispersion minimizing finite difference methods. We present both a strong scalability study and a complexity analysis performed on a massively parallel distributed memory computer. Numerical results demonstrate the usefulness of the algorithm on a realistic three-dimensional application at high frequency.

Keywords

Finite Difference Scheme Multigrid Method Perfectly Match Layer Acoustic Imaging Krylov Subspace Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors would like to thank TOTAL for the financial support over the past years. They also would like to acknowledge GENCI (Grand Equipement National de Calcul Intensif) for the dotation of computing hours on the IBM BG/Q computer at IDRIS, France. This work was granted access to the HPC resources of IDRIS under allocation 2015065068 and 2016065068 made by GENCI.

References

  1. 1.
    P. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J.-Y. L’Excellent, and C. Weisbecker. Improving multifrontal methods by means of block low-rank representations. SIAM J. Sci. Comput., 37(3):A1451–A1474, 2015.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    F. Aminzadeh, J. Brac, and T. Kunz. 3D Salt and Overthrust models. SEG/EAGE modeling series I, Society of Exploration Geophysicists, 1997.Google Scholar
  3. 3.
    A. Bayliss, C. I. Goldstein, and E. Turkel. An iterative method for the Helmholtz equation. J. Comp. Phys., 49:443–457, 1983.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J.-P. Berenger. A perfectly matched layer for absorption of electromagnetic waves. J. Comp. Phys., 114:185–200, 1994.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    J.-P. Berenger. Three-dimensional perfectly matched layer for absorption of electromagnetic waves. J. Comp. Phys., 127:363–379, 1996.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    M. Bollhöfer, M. J. Grote, and O. Schenk. Algebraic multilevel preconditioner for the solution of the Helmholtz equation in heterogeneous media. SIAM J. Sci. Comput., 31:3781–3805, 2009.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    A. Brandt and I. Livshits. Wave-ray multigrid method for standing wave equations. Electron. Trans. Numer. Anal., 6:162–181, 1997.MathSciNetMATHGoogle Scholar
  8. 8.
    H. Calandra, S. Gratton, R. Lago, X. Vasseur, and L. M. Carvalho. A modified block flexible GMRES method with deflation at each iteration for the solution of non-hermitian linear systems with multiple right-hand sides. SIAM J. Sci. Comput., 35(5):S345–S367, 2013.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    H. Calandra, S. Gratton, R. Lago, X. Pinel, and X. Vasseur. Two-level preconditioned Krylov subspace methods for the solution of three-dimensional heterogeneous Helmholtz problems in seismics. Numerical Analysis and Applications, 5:175–181, 2012.CrossRefMATHGoogle Scholar
  10. 10.
    H. Calandra, S. Gratton, J. Langou, X. Pinel, and X. Vasseur. Flexible variants of block restarted GMRES methods with application to geophysics. SIAM J. Sci. Comput., 34(2):A714–A736, 2012.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    H. Calandra, S. Gratton, X. Pinel, and X. Vasseur. An improved two-grid preconditioner for the solution of three-dimensional Helmholtz problems in heterogeneous media. Numer. Linear Algebra Appl., 20, pp. 663–688, 2013.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Z. Chen, D. Cheng, and T. Wu. A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation. J. Comp. Phys., 231:8152–8175, 2012.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    G. Cohen. Higher-order numerical methods for transient wave equations. Springer, 2002.CrossRefMATHGoogle Scholar
  14. 14.
    Y. Diouane, S. Gratton, X. Vasseur, L. N. Vicente, and H. Calandra. A parallel evolution strategy for an Earth imaging problem in geophysics. Optimization and Engineering, 17(1):3–26, 2016.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    H. Elman, O. Ernst, D. O’Leary, and M. Stewart. Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering. Comput. Methods Appl. Mech. Engrg., 194(1):1037–1055, 2005.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    H. C. Elman, O. G. Ernst, and D. P. O’Leary. A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations. SIAM J. Sci. Comput., 23:1291–1315, 2001.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Modeling and Simulation, 9:686–710, 2011.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation. Comm. Pure Appl. Math., 64:697–735, 2011.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Y. A. Erlangga. A robust and efficient iterative method for the numerical solution of the Helmholtz equation. PhD thesis, TU Delft, 2005.Google Scholar
  20. 20.
    Y. A. Erlangga. Advances in iterative methods and preconditioners for the Helmholtz equation. Archives of Computational Methods in Engineering, 15:37–66, 2008.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Y. A. Erlangga and R. Nabben. On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian. Electron. Trans. Numer. Anal., 31:403–424, 2008.MathSciNetMATHGoogle Scholar
  22. 22.
    Y. A. Erlangga, C. Oosterlee, and C. Vuik. A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput., 27:1471–1492, 2006.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Y. A. Erlangga, C. Vuik, and C. Oosterlee. On a class of preconditioners for solving the Helmholtz equation. Appl. Num. Math., 50:409–425, 2004.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    O. Ernst and M. J. Gander. Why it is difficult to solve Helmholtz problems with classical iterative methods. In O. Lakkis I. Graham, T. Hou and R. Scheichl, editors, Numerical Analysis of Multiscale Problems. Springer, 2011.Google Scholar
  25. 25.
    C. Farhat, A. Macedo, and M. Lesoinne. A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems. Numer. Math., 85:283–308, 2000.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    M. Gander, I. G. Graham, and E. A. Spence. Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed? Numer. Math., 131:567–614, 2015.MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    W. Gropp, E. Lusk, and A. Skjellum. Using MPI: Portable Parallel Programming with the Message-Passing Interface. MIT Press, 1999.MATHGoogle Scholar
  28. 28.
    W. Hackbusch and U. Trottenberg. Multigrid methods. Springer, 1982. Lecture Notes in Mathematics, vol. 960, Proceedings of the conference held at Köln-Porz, November 23–27 1981.Google Scholar
  29. 29.
    I. Harari and E. Turkel. Accurate finite difference methods for time-harmonic wave propagation. J. Comp. Phys., 119:252–270, 1995.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    M. Hoemmen. Communication-avoiding Krylov subspace methods. PhD thesis, University of California, Berkeley, Department of Computer Science, 2010.Google Scholar
  31. 31.
    F. Liu and L. Ying Additive sweeping preconditioner for the Helmholtz equation ArXiv e-prints, 2015. http://arxiv.org/abs/1504.04058.
  32. 32.
    F. Liu and L. Ying Recursive sweeping preconditioner for the 3D Helmholtz equation ArXiv e-prints, 2015. http://arxiv.org/abs/1502.07266.
  33. 33.
    S. P. MacLachlan and C. W. Oosterlee. Algebraic multigrid solvers for complex-valued matrices. SIAM J. Sci. Comput., 30:1548–1571, 2008.MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    S. Operto, J. Virieux, P. R. Amestoy, J.-Y. L’Excellent, L. Giraud, and H. Ben Hadj Ali. 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study. Geophysics, 72–5:195–211, 2007.Google Scholar
  35. 35.
    J. Poulson, B. Engquist, S. Li, and L. Ying. A parallel sweeping preconditioner for heterogeneous 3d Helmholtz equations. SIAM J. Sci. Comput., 35:C194–C212, 2013.MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    X. Pinel. A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics. PhD thesis, CERFACS, 2010. TH/PA/10/55.Google Scholar
  37. 37.
    B. Reps, W. Vanroose, and H. bin Zubair. On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning. J. Comp. Phys., 229:8384–8405, 2010.Google Scholar
  38. 38.
    C. D. Riyanti, A. Kononov, Y. A. Erlangga, R.-E. Plessix, W. A. Mulder, C. Vuik, and C. Oosterlee. A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation. J. Comp. Phys., 224:431–448, 2007.MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Y. Saad. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Scientific and Statistical Computing, 14:461–469, 1993.MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, 2003. Second edition.Google Scholar
  41. 41.
    Y. Saad and M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Scientific and Statistical Computing, 7:856–869, 1986.MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    A.H. Sheikh, D. Lahaye, and C. Vuik. On the convergence of shifted Laplace preconditioner combined with multilevel deflation. Numer. Linear Algebra Appl., 20:645–662, 2013.MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    V. Simoncini and D. B. Szyld. Flexible inner-outer Krylov subspace methods. SIAM J. Numer. Anal., 40:2219–2239, 2003.MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    V. Simoncini and D. B. Szyld. Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl., 14:1–59, 2007.MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    F. Sourbier, S. Operto, J. Virieux, P. Amestoy, and J. Y. L’ Excellent. FWT2D: a massively parallel program for frequency-domain full-waveform tomography of wide-aperture seismic data - part 1: algorithm. Computer & Geosciences, 35:487–495, 2009.Google Scholar
  46. 46.
    F. Sourbier, S. Operto, J. Virieux, P. Amestoy, and J. Y. L’ Excellent. FWT2D: a massively parallel program for frequency-domain full-waveform tomography of wide-aperture seismic data - part 2: numerical examples and scalability analysis. Computer & Geosciences, 35:496–514, 2009.Google Scholar
  47. 47.
    C. Stolk, M. Ahmed, and S. K. Bhowmik. A multigrid method for the Helmholtz equation with optimized coarse grid correction. SIAM J. Sci. Comput., 36:A2819–A2841, 2014.MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    C. Stolk. A rapidly converging domain decomposition method for the Helmholtz equation. J. Comp. Phys., 241:240–252, 2013.CrossRefGoogle Scholar
  49. 49.
    K. Stüben and U. Trottenberg. Multigrid methods: fundamental algorithms, model problem analysis and applications. In W. Hackbusch and U. Trottenberg, editors, Multigrid methods, Koeln-Porz, 1981, Lecture Notes in Mathematics, volume 960. Springer, 1982.Google Scholar
  50. 50.
    A. Tarantola. Inverse problem theory and methods for model parameter estimation. SIAM, 2005.CrossRefMATHGoogle Scholar
  51. 51.
    E. Turkel, D. Gordon, R. Gordon, and S. Tsynkov. Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wavenumber. J. Comp. Phys., 232:272–287, 2013.CrossRefMATHGoogle Scholar
  52. 52.
    A. Toselli and O. Widlund. Domain Decomposition methods - Algorithms and Theory. Springer Series on Computational Mathematics, Springer, 34, 2005.Google Scholar
  53. 53.
    U. Trottenberg, C. W. Oosterlee, and A. Schüller. Multigrid. Academic Press Inc., 2001.MATHGoogle Scholar
  54. 54.
    N. Umetani, S. P. MacLachlan, and C. W. Oosterlee. A multigrid-based shifted Laplacian preconditioner for fourth-order Helmholtz discretization. Numer. Linear Algebra Appl., 16:603–626, 2009.MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    M. B. van Gijzen, Y. A. Erlangga, and C. Vuik. Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput., 29:1942–1958, 2007.MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    W. Vanroose, B. Reps, and H. bin Zubair. A polynomial multigrid smoother for the iterative solution of the heterogeneous Helmholtz problem. Technical Report, University of Antwerp, Belgium, 2010. http://arxiv.org/abs/1012.5379.
  57. 57.
    P. S. Vassilevski. Multilevel Block Factorization Preconditioners, Matrix-based Analysis and Algorithms for Solving Finite Element Equations. Springer, New York, 2008.MATHGoogle Scholar
  58. 58.
    J. Virieux and S. Operto. An overview of full waveform inversion in exploration geophysics. Geophysics, 74(6):WCC127–WCC152, 2009.CrossRefGoogle Scholar
  59. 59.
    S. Wang, M. V. de Hoop, and J. Xia. Acoustic inverse scattering via Helmholtz operator factorization and optimization. J. Comp. Phys., 229:8445–8462, 2010.MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    S. Wang, M. V. de Hoop, and J. Xia. On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver. Geophysical Prospecting, 59:857–873, 2011.CrossRefGoogle Scholar
  61. 61.
    L. Zepeda-Núñez and L. Demanet. The method of polarized traces for the 2D Helmholtz equation. J. Comp. Phys., 308:347–388, 2016.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.TOTAL E&P Research and Technology USAHoustonUSA
  2. 2.INPT-IRITUniversity of Toulouse and ENSEEIHTToulouse Cedex 7France
  3. 3.ISAE-SUPAEROToulouse Cedex 4France

Personalised recommendations