Solution of the Helmholtz problem using the preconditioned low-rank approximation technique


  • K.V. Voronin The Institute of Computational Mathematics and Mathematical Geophysics of SB RAS (ICM&MG SB RAS)
  • S.A. Solovyev Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS



3D Helmholtz problem, algorithms for sparse systems of linear algebraic equations, Gaussian elimination method, low-rank approximation, HSS matrix representation, BiCGStab method, iterative refinement


An algorithm for solving the Helmholtz problem in 3D heterogeneous media using the low-rank approximation technique is proposed. This technique is applied as a preconditioner for two different iterative processes: an iterative refinement and BiCGStab. The iterative refinement approach is known to be very simple and straightforward but can suffer from the lack of convergence; BiCGStab is more stable and more sophisticated as well. A dependence of the convergence rate on low-rank approximation quality is studied for these iterative processes. For typical problems of seismic exploration, it is shown that, starting with some low-rank accuracy, the convergence rate of the iterative refinement is very similar to BiCGStab. Therefore, it is preferable to use the more efficient iterative refinement method. Numerical experiments also show that, for reasonable (from the practical standpoint) low-rank accuracy, the proposed method provides three times performance gain (for sequential code) and reduces the memory usage up to a factor of two in comparison with the Intel MKL PARDISO high performance direct solver.

Author Biographies

K.V. Voronin

S.A. Solovyev


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How to Cite

Воронин К., Соловьев С. Solution of the Helmholtz Problem Using the Preconditioned Low-Rank Approximation Technique // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2015. 16. 268-280. doi 10.26089/NumMet.v16r226



Section 1. Numerical methods and applications