Parallel Computations for Solving 3D Helmholtz Problem by Using Direct Solver with Low-Rank Approximation and HSS Technique

  • Boris Glinskiy
  • Nikolay Kuchin
  • Victor Kostin
  • Sergey SolovyevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)


The modern methods of processing the geophysical data, such as Reverse Time Migration (RTM) and Full Waveform Inversion (FWI) require solving series of forward problems where the main step is solution of Systems of Linear Algebraic Equations (SLAE) of big size. For big sizes, it is time and memory consuming problem.

In this paper, we present a parallel direct algorithm to solve boundary value problems for 3D Helmholtz equation discretized with help of finite differences. The memory consumption has been resolved due to Nested Dissection approach, low-rank approximation technique and HSS format. OpenMP parallelization is based on standard BLAS and LAPACK functionality. For MPI parallelization, we propose a novel algorithm that uses dynamical distribution of the elimination tree nodes across cluster nodes. Numerical experiments show performance benefits of the proposed cluster algorithm compared to the not parallel version and demonstrate significant memory advantages over direct solvers without low-rank approximation.


Helmholtz Equation Perfectly Match Layer Diagonal Block Cluster Node Direct Solver 
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The research described was partially supported by RFBR grants 14-05-00049, 14-05-93090, 16-05-00800 and the Russian Academy of Sciences Program “Arctic”. All computations were performed on cluster NSK-30T of Siberian Supercomputer Center.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Boris Glinskiy
    • 1
  • Nikolay Kuchin
    • 1
  • Victor Kostin
    • 2
  • Sergey Solovyev
    • 2
    Email author
  1. 1.Siberian Supercomputer CenterInstitute of Computational Mathematics and Mathematical Geophysics SB RASNovosibirskRussia
  2. 2.Institute of Petroleum Geology and Geophysics SB RASNovosibirskRussia

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