Advertisement

Computational Geosciences

, Volume 20, Issue 2, pp 297–315 | Cite as

Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units

  • H. KnibbeEmail author
  • C. Vuik
  • C. W. Oosterlee
Open Access
ORIGINAL PAPER
First Online:

Abstract

In geophysical applications, the interest in least-squares migration (LSM) as an imaging algorithm is increasing due to the demand for more accurate solutions and the development of high-performance computing. The computational engine of LSM in this work is the numerical solution of the 3D Helmholtz equation in the frequency domain. The Helmholtz solver is Bi-CGSTAB preconditioned with the shifted Laplace matrix-dependent multigrid method. In this paper, an efficient LSM algorithm is presented using several enhancements. First of all, a frequency decimation approach is introduced that makes use of redundant information present in the data. It leads to a speedup of LSM, whereas the impact on accuracy is kept minimal. Secondly, a new matrix storage format Very Compressed Row Storage (VCRS) is presented. It not only reduces the size of the stored matrix by a certain factor but also increases the efficiency of the matrix-vector computations. The effects of lossless and lossy compression with a proper choice of the compression parameters are positive. Thirdly, we accelerate the LSM engine by graphics cards (GPUs). A GPU is used as an accelerator, where the data is partially transferred to a GPU to execute a set of operations or as a replacement, where the complete data is stored in the GPU memory. We demonstrate that using the GPU as a replacement leads to higher speedups and allows us to solve larger problem sizes. Summarizing the effects of each improvement, the resulting speedup can be at least an order of magnitude compared to the original LSM method.

Keywords

Least-squares migration Helmholtz equation Wave equation Frequency domain Multigrid method GPU acceleration Matrix storage format Frequency decimation 

Mathematics Subject Classifications (2010)

65-04 65N55 86A15 65Y05 
Download to read the full article text

References

  1. 1.
    Aminzadeh, F., Brac, J., Kunz, T.: 3-D Salt and Overthrust Models. Society of Exploration Geophysicists, Tulsa (1997)Google Scholar
  2. 2.
    Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H.: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  3. 3.
    Brackenridge, K.: Multigrid and cyclic reduction applied to the Helmholtz equation. In: Melson, N.D., Manteufel, T.A., McCormick, S.F. (eds.) 6th Cooper Mountain Conf. on Multigrid Methods, pp. 31–41 (1993)Google Scholar
  4. 4.
    Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. Society for Industrial and Applied Mathematics, Philadelphia (2006). http://opac.inria.fr/record=b1120110 CrossRefGoogle Scholar
  5. 5.
    Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Math. Comput. 31, 629–651 (1977)CrossRefGoogle Scholar
  6. 6.
    Erlangga, Y.A., Vuik, C., Oosterlee, C.W.: On a class of preconditioners for solving the discrete Helmholtz equation. In: Cohen, G., Heikkola, E., Joly, P., Neittaanmakki, P. (eds.) Mathematical and Numerical Aspects of Wave Propagation, pp. 788–793. Univ Jyväskylä, Finnland (2003)Google Scholar
  7. 7.
    Erlangga, Y.A., Oosterlee, C.W., Vuik, C.: A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27, 1471–1492 (2006)CrossRefGoogle Scholar
  8. 8.
    Gersho, A., Grey, R.M.: Vector Quantization and Signal Compression. Springer Science+Business Media (1992). doi: 10.1007/978-1-4615-3626- Google Scholar
  9. 9.
    van Gijzen, M.B., Erlangga, Y.A., Vuik, C.: Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplace. SIAM J. Sci. Comput. 29, 1942–1958 (2007)CrossRefGoogle Scholar
  10. 10.
    Gozani, J., Nachshon, A., Turkel, E.: Conjugate gradient coupled with multigrid for an indefinite problem. In: Vichnevestsky R, Tepelman, R. S. (eds.) Advances in Computational Methods for PDEs V, pp. 425–427. IMACS, New Brunswick (1984)Google Scholar
  11. 11.
    Guitton, A., Diaz, E.: Attenuating crosstalk noise with simultaneous source full waveform inversion. Geophys. Prosp. 60, 759–768 (2012). doi: 10.1111/j.1365-2478.2011.01023.x CrossRefGoogle Scholar
  12. 12.
    Kechroud, R., Soulaimani, A., Saad, Y., Gowda, S.: Preconditioning techniques for the solution of the Helmholtz equation by the finite element method. Math. Comput. Simul. 65(4-5), 303–321 (2004). doi: 10.1016/j.matcom.2004.01.004 CrossRefGoogle Scholar
  13. 13.
    Khronos Group (2014) www.khronos.org
  14. 14.
    Kim, Y., Min, D.J., Shin, C.: Frequency-domain reverse-time migration with source estimation. Geophysics 76(2), S41–S49 (2011)CrossRefGoogle Scholar
  15. 15.
    Knibbe, H., Oosterlee, C.W., Vuik, C.: GPU implementation of a Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method. J. Comput. Appl. Math. 236, 281–293 (2011). doi: 10.1016/j.cam.2011.07.021 CrossRefGoogle Scholar
  16. 16.
    Knibbe, H., Vuik, C., Oosterlee, C.W.: 3D Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method on multi-GPUs. In: Cangiani, A, Davidchack, R.L., Georgoulis, E, Gorban, A.N., Levesley, J, Tretyakov, M.V. (eds.) Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, pp 653–661. Springer-Verlag, Berlin Heidelberg (2011)Google Scholar
  17. 17.
    Knibbe, H., Mulder, W.A., Oosterlee, C.W., Vuik, C.: Closing the performance gap between an iterative frequency-domain solver and an explicit time-domain scheme for 3-d migration on parallel architectures. Geophysics 79, 47–61 (2014)CrossRefGoogle Scholar
  18. 18.
    Kourtis, K., Goumas, G., Koziris, N.: Optimizing sparse matrix-vector multiplication using index and value compression. In: Proceedings of the 5th Conference on Computing Frontiers CF ’08, pp. 87–96. ACM, New York (2008)Google Scholar
  19. 19.
    Laird, A.L., Giles, M.B.: Preconditioned iterative solution of the 2D Helmholtz equation. Tech. Rep. 02/12, Oxford Computing Laboratory, Oxford, UK (2002)Google Scholar
  20. 20.
    LGM (2012) The Little Green Machine: Massive many-core supercomputer at low environmental cost. http://www.littlegreenmachine.org
  21. 21.
    Mulder, W.A., Plessix, R.E.: How to choose a subset of frequencies in frequency-domain finite-difference migration. Geophys. J. Int. 158(3), 801–812 (2004). doi: 10.1111/j.1365-246X.2004.02336.x CrossRefGoogle Scholar
  22. 22.
    Nemeth, T., Wu, C., Schuster, G.T.: Least-squares migration of incomplete reflection data. Geophysics 64(1), 208–221 (1999)CrossRefGoogle Scholar
  23. 23.
    Plessix, R.E., Mulder, W.A.: Frequency-domain finite-difference amplitude-preserving migration. Geophys. J. Int. 157, 975–987 (2004)CrossRefGoogle Scholar
  24. 24.
    Ren, H., Wang, H., Chen, S.: Least-squares reverse time migration in frequency domain using the adjoint-state method. J. Geophys. Eng. 10(3), 035, 002 (2013) http://stacks.iop.org/1742-2140/10/i=3/a=035002 CrossRefGoogle Scholar
  25. 25.
    Riyanti, C.D., Kononov, A., Erlangga, Y.A., Vuik, C., Oosterlee, C.W., Plessix, R.E., Mulder, W.A.: A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation. J. Comput. Phys. 224(1), 431–448 (2007). doi: 10.1016/j.jcp.2007.03.033 CrossRefGoogle Scholar
  26. 26.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)CrossRefGoogle Scholar
  27. 27.
    Schuster, G.T.: Least-squares crosswell migration. In: SEG Expanded Abstracts 12, 63 Annual International Meeting, pp. 25–28 (1993)Google Scholar
  28. 28.
    Stüben, K., Trottenberg, U.: Multigrid methods: Fundamental algorithms, model problem analysis and applications. In: Hackbush, W., Trottenberg, U. (eds.) Lecture Notes in Math, vol. 960, pp. 1–176 (1982)Google Scholar
  29. 29.
    Tang, Y.: Wave-equation Hessian by phase encoding. In: 78 Annual International Meeting, SEG, Expanded Abstracts, vol. 27, pp. 2201–2205 (2008)Google Scholar
  30. 30.
    Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, New York (2001)Google Scholar
  31. 31.
    Turkel, E.: Numerical methods and nature. J. Sci. Comput. 28, 549–570 (2006)CrossRefGoogle Scholar
  32. 32.
    Wei, D., Schuster, G.T.: Least-squares migration of multisource data with a deblurring filter. Geophysics 76(5), R135–R146 (2011)Google Scholar
  33. 33.
    Wienands, R., Oosterlee, C.W.: On three-grid Fourier analysis of multigrid. SIAM J. Sci. Comp. 23, 651–671 (2001)CrossRefGoogle Scholar
  34. 34.
    Zhebel, E.: A multigrid method with matrix-dependent transfer operators for 3D diffusion problems with jump coefficients, PhD thesis, Technical University Bergakademie Freiberg, Germany (2006)Google Scholar

Copyright information

© The Author(s) 2015

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.Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  2. 2.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands

Personalised recommendations