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


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.


  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)

    Book  Google 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)

  4. 4.

    Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. Society for Industrial and Applied Mathematics, Philadelphia (2006).

    Book  Google Scholar 

  5. 5.

    Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Math. Comput. 31, 629–651 (1977)

    Article  Google 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)

  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)

    Article  Google 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)

    Article  Google 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)

  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

    Article  Google 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

    Article  Google Scholar 

  13. 13.

    Khronos Group (2014)

  14. 14.

    Kim, Y., Min, D.J., Shin, C.: Frequency-domain reverse-time migration with source estimation. Geophysics 76(2), S41–S49 (2011)

    Article  Google 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/

    Article  Google 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)

  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)

    Article  Google 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)

  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)

  20. 20.

    LGM (2012) The Little Green Machine: Massive many-core supercomputer at low environmental cost.

  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

    Article  Google Scholar 

  22. 22.

    Nemeth, T., Wu, C., Schuster, G.T.: Least-squares migration of incomplete reflection data. Geophysics 64(1), 208–221 (1999)

    Article  Google Scholar 

  23. 23.

    Plessix, R.E., Mulder, W.A.: Frequency-domain finite-difference amplitude-preserving migration. Geophys. J. Int. 157, 975–987 (2004)

    Article  Google 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)

    Article  Google 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/

    Article  Google Scholar 

  26. 26.

    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  Google Scholar 

  27. 27.

    Schuster, G.T.: Least-squares crosswell migration. In: SEG Expanded Abstracts 12, 63 Annual International Meeting, pp. 25–28 (1993)

  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)

  29. 29.

    Tang, Y.: Wave-equation Hessian by phase encoding. In: 78 Annual International Meeting, SEG, Expanded Abstracts, vol. 27, pp. 2201–2205 (2008)

  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)

    Article  Google Scholar 

  32. 32.

    Wei, D., Schuster, G.T.: Least-squares migration of multisource data with a deblurring filter. Geophysics 76(5), R135–R146 (2011)

  33. 33.

    Wienands, R., Oosterlee, C.W.: On three-grid Fourier analysis of multigrid. SIAM J. Sci. Comp. 23, 651–671 (2001)

    Article  Google 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)

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Correspondence to H. Knibbe.

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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, 297–315 (2016).

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  • 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