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Parallel multilevel recursive approximate inverse techniques for solving general sparse linear systems

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Abstract

In this article, a new parallel multilevel algebraic recursive generic approximate inverse solver (PMARGAIS) is proposed. PMARGAIS utilizes the parallel modified generic factored approximate sparse inverse (PMGenFAspI) matrix technique designed for shared memory parallel systems. PMARGAIS requires a block independent set reordering scheme, to create a hierarchy of levels. A modified block breadth first search (MBBFS) is proposed for reducing memory requirements and retaining load balancing. The SVD method is used to compute the inverse of the independent blocks that are formed from the reordering scheme, and computes accurately the Schur complement that is used as a coefficient matrix on the next level, resulting in a hybrid direct-iterative method for large linear systems. The solution of the linear system at the last level is performed with the parallel explicit preconditioned BiCGSTAB method in conjunction with the PMGenFAspI matrix. The parallelization of the proposed methods uses the vector units of modern CPUs. Implementation details are provided and numerical results are given demonstrating the applicability and effectiveness of the proposed schemes.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, School of Engineering, Democritus University of Thrace, University Campus, Kimmeria, 67100, Xanthi, Greece

    Antonios T. Makaratzis, Christos K. Filelis-Papadopoulos & George A. Gravvanis

Authors
  1. Antonios T. Makaratzis
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  2. Christos K. Filelis-Papadopoulos
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  3. George A. Gravvanis
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Corresponding author

Correspondence to George A. Gravvanis.

Appendix

Appendix

The PEPBiCGSTAB method, using AVX units, is described by the following algorithmic scheme:

figure f
figure g

where fmadd(xr3, xr1, xr2) is the fused multiply add operation \({xr3}={xr3}+{xr1}*{xr2}\), where xr1, xr2 and xr3 are vectors consisting of four double-precision floating point numbers.

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Makaratzis, A.T., Filelis-Papadopoulos, C.K. & Gravvanis, G.A. Parallel multilevel recursive approximate inverse techniques for solving general sparse linear systems. J Supercomput 72, 2259–2282 (2016). https://doi.org/10.1007/s11227-016-1728-5

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  • DOI: https://doi.org/10.1007/s11227-016-1728-5

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