Abstract
It was recently shown that block-circulant preconditioners applied to a conjugate gradient method used to solve structured sparse linear systems arising from 2D or 3D elliptic problems have good numerical properties and a potential for high parallel efficiency. The asymptotic estimate for their convergence rate is as for the incomplete factorization methods but the efficiency of the parallel algorithms based on circulant preconditioners are asymptotically optimal. In this paper parallel performance of a circulant block-factorization based preconditioner applied to a 3D model problem is investigated. The aim of this presentation is to analyze the performance and to report on the experimental results obtained on shared and distributed memory parallel architectures. A portable parallel code is developed based on Message Passing Interface (MPI) and OpenMP (Open Multi Processing) standards. The performed numerical tests on a wide range of parallel computer systems clearly demonstrate the high level of parallel efficiency of the developed parallel code.
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References
Chan, R.H., Chan, T.F.: Circulant preconditioners for elliptic problems. J. Num. Lin. Alg. Appl. 1, 77–101 (1992)
Lirkov, I., Margenov, S.: Conjugate gradient method with circulant block-factorization preconditioners for 3D elliptic problems. In: Proceedings of the Workshop # 3 Copernicus 94-0820 “HIPERGEOS” project meeting, Rožnov pod Radhoštěm (1996)
Lirkov, I., Margenov, S.: Parallel complexity of conjugate gradient method with circulant block-factorization preconditioners for 3D elliptic problems. In: Iliev, O.P., Kaschiev, M.S., Sendov, B.l., Vassilevski, P.V. (eds.) Recent Advances in Numerical Methods and Applications, pp. 455–463. World Scientific, Singapore (1999)
Lirkov, I., Margenov, S., Vassilevski, P.S.: Circulant block-factorization preconditioners for elliptic problems. Computing 53(1), 59–74 (1994)
Lirkov, I., Margenov, S., Zikatanov, L.: Circulant block–factorization preconditioning of anisotropic elliptic problems. Computing 58(3), 245–258 (1997)
Van Loan, C.: Computational frameworks for the fast Fourier transform. SIAM, Philadelphia (1992)
Saad, Y., Schultz, M.H.: Data communication in parallel architectures. Parallel Comput. 11, 131–150 (1989)
Snir, M., Otto, S., Huss-Lederman, S., Walker, D., Dongara, J.: MPI: The Complete Reference. Scientific and engineering computation series. The MIT Press, Cambridge (1997) (Second printing)
Walker, D., Dongara, J.: MPI: a standard Message Passing Interface. Supercomputer 63, 56–68 (1996)
Web-site for Open MP tools, http://www.openmp.org
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© 2005 Springer-Verlag Berlin Heidelberg
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Lirkov, I. (2005). Parallel Performance of a 3D Elliptic Solver. In: Li, Z., Vulkov, L., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2004. Lecture Notes in Computer Science, vol 3401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31852-1_46
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DOI: https://doi.org/10.1007/978-3-540-31852-1_46
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