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


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.


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 


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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

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