Geodesic
Aspects of parallel computing to solve Helmholtz equation by a direct solver with low-rank approximation and the HSS format of data storage
Numerical methods and programming, Tome 16 (2015) no. 4, pp. 607-616.

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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. 
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     author = {B. M. Glinskii and V. I. Kostin and N. V. Kuchin and S. A. Solovyev and V. A. Tcheverda},
     title = {Aspects of parallel computing to solve {Helmholtz} equation by a direct solver with low-rank approximation and the {HSS} format of data storage},
     journal = {Numerical methods and programming},
     pages = {607--616},
     publisher = {mathdoc},
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     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2015_16_4_a14/}
}
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B. M. Glinskii; V. I. Kostin; N. V. Kuchin; S. A. Solovyev; V. A. Tcheverda. Aspects of parallel computing to solve Helmholtz equation by a direct solver with low-rank approximation and the HSS format of data storage. Numerical methods and programming, Tome 16 (2015) no. 4, pp. 607-616. http://geodesic.mathdoc.fr/item/VMP_2015_16_4_a14/