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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aminzadeh, F., Brac, J., Kunz, T.: 3-D Salt and Overthrust Models. Society of Exploration Geophysicists, Tulsa (1997)
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)
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)
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
Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Math. Comput. 31, 629–651 (1977)
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)
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)
Gersho, A., Grey, R.M.: Vector Quantization and Signal Compression. Springer Science+Business Media (1992). doi:10.1007/978-1-4615-3626-
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)
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)
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
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
Khronos Group (2014) www.khronos.org
Kim, Y., Min, D.J., Shin, C.: Frequency-domain reverse-time migration with source estimation. Geophysics 76(2), S41–S49 (2011)
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
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)
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)
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)
Laird, A.L., Giles, M.B.: Preconditioned iterative solution of the 2D Helmholtz equation. Tech. Rep. 02/12, Oxford Computing Laboratory, Oxford, UK (2002)
LGM (2012) The Little Green Machine: Massive many-core supercomputer at low environmental cost. http://www.littlegreenmachine.org
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
Nemeth, T., Wu, C., Schuster, G.T.: Least-squares migration of incomplete reflection data. Geophysics 64(1), 208–221 (1999)
Plessix, R.E., Mulder, W.A.: Frequency-domain finite-difference amplitude-preserving migration. Geophys. J. Int. 157, 975–987 (2004)
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
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
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Schuster, G.T.: Least-squares crosswell migration. In: SEG Expanded Abstracts 12, 63 Annual International Meeting, pp. 25–28 (1993)
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)
Tang, Y.: Wave-equation Hessian by phase encoding. In: 78 Annual International Meeting, SEG, Expanded Abstracts, vol. 27, pp. 2201–2205 (2008)
Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, New York (2001)
Turkel, E.: Numerical methods and nature. J. Sci. Comput. 28, 549–570 (2006)
Wei, D., Schuster, G.T.: Least-squares migration of multisource data with a deblurring filter. Geophysics 76(5), R135–R146 (2011)
Wienands, R., Oosterlee, C.W.: On three-grid Fourier analysis of multigrid. SIAM J. Sci. Comp. 23, 651–671 (2001)
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)
Author information
Authors and Affiliations
Corresponding author
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.
About this article
Cite this article
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). https://doi.org/10.1007/s10596-015-9546-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-015-9546-z
Keywords
- Least-squares migration
- Helmholtz equation
- Wave equation
- Frequency domain
- Multigrid method
- GPU acceleration
- Matrix storage format
- Frequency decimation