"Aspects of parallel computing to solve Helmholtz equation by a direct solver with low-rank approximation and the HSS format of data storage"
Glinskiy B.M., Kostin V.I., Kuchin N.V., Solovyev S.A., and Cheverda V.A.

An algorithm for solving systems of linear algebraic equations based on the Gaussian elimination method is proposed. The algorithm is aimed to solve boundary value problems for the Helmholtz equation in 3D heterogeneous media. In order to solve linear systems raised from geophysical applications, we developed a parallel version targeted on heterogeneous high-performance computing clusters (MPP and SMP architecture). Using the low-rank approximation technique and the HSS format allows us to solve problems larger than by the use of traditional direct solvers with saving the L-factor in full rank (FR). Using the proposed approach reduces computation time; it is the key-point of 3D geophysical problems. Numerical experiments demonstrate a number of advantages of the proposed low-rank approach in comparison with direct solvers (FR-approaches).

Keywords: Helmholtz equation, algorithms for sparse systems of linear algebraic equations, Gaussian elimination method, low-rank approximation, HSS matrix representation, distributed parallel systems, heterogeneous high-performance computing systems.

  • Glinskiy B.M. – Institute of Computational Mathematics and Mathematical Geophysics, Siberian Supercomputer Center, Siberian Branch of Russian Academy of Sciences; prospekt Lavrentyeva 6, Novosibirsk, 630090, Russia; Dr. Sci., Professor, Executive Director, e-mail: gbm@sscc.ru
  • Kostin V.I. – Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of Russian Academy of Sciences; prospekt Koptyuga 3, Novosibirsk, 630090, Russia; Ph.D., Senior Scientist, e-mail: vctrkstn@mail.ru
  • Kuchin N.V. – Institute of Computational Mathematics and Mathematical Geophysics, Siberian Supercomputer Center, Siberian Branch of Russian Academy of Sciences; prospekt Lavrentyeva 6, Novosibirsk, 630090, Russia; Senior Specialist, e-mail: kuchin@sscc.ru
  • Solovyev S.A. – Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of Russian Academy of Sciences; prospekt Koptyuga 3, Novosibirsk, 630090, Russia; Ph.D., Scientist, e-mail: 511ssa@mail.ru
  • Cheverda V.A. – Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of Russian Academy of Sciences; prospekt Koptyuga 3, Novosibirsk, 630090, Russia; Dr. Sci., Head of Laboratory, e-mail: CheverdaVA@ipgg.sbras.ru