Abstract
In this paper we study the parallel aspects of IMGS, Incomplete Modified Gram-Schmidt preconditioner which can be used for efficiently solving sparse and large linear systems and least squares problems on massively parallel distributed memory computers. The performance of this preconditioning technique on this kind of architecture is always limited because of the global communication required for the inner products, even for ParIMGS, a parallel version of IMGS where we create some possibilities such that the computation can be overlapped with the communication. We will describe a more efficient alternative, namely Improved ParIMGS (IParIMGS) which avoids the global communication of inner products and only requires local communications. Therefore, the cost of communication can be significantly reduced. Several numerical experiments carried out on Parsytec GC/PowerPlus are presented as well.
References
S. Ashby. Polynomial preconditioning for conjugate gradient methods. PhD thesis, Department of Computer Science, University of Illinois Urbana-Champaign, 1987.
Z. Bai, D. Hu, and L. Reichel. A newton basis GMRES implementation. Technical Report 91-03, University of Kentucky, 1991.
Å. Björck and T. Elfving. Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations. BIT, 19:145–163, 1979.
L. G. C. Crone and H. A. van der Vorst. Communication aspects of the conjugate gradient method on distributed memory machines. Supercomputer, X(6):4–9, 1993.
E. de Sturler. A parallel variant of the GMRES(m). In Proceedings of the 13th IMACS World Congress on Computational and Applied Mathematics. IMACS. Criterion Press, 1991.
E. de Sturler and H. A. van der Vorst. Reducing the effect of the global communication in GMRES(m) and CG on parallel distributed memory computers. Technical Report 832, Mathematical Institute, University of Utrecht, Utrecht, The Netheland, 1994.
A. Jennings and M. A. Ajiz. Incomplete methods for solving A T Aχ = b. SIAM Journal on Scientific and Statisticgal Computing, 5:978–987, 1984.
J. A. Meijerink and H. A. van der Vorst. An iterative solution method for linear system of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation, 31(137):148–162, 1977.
X. Wang. Incomplete factorization preconditioning for linear least squares problems. PhD thesis, Department of Computer Science, University of Illinois at Urbana-Champaign, 1993.
T. Yang. Error analysis for incomplete modified Gram-Schmidt preconditioner. In Proceedings of Prague Mathematical Conference (PMC-96), July 1996. Mathematical Institute of the Academy of Sciences, Zitza 25, CZ-115 67 Praha, Czech Republic.
T. Yang. Solving sparse least squares problems on massively parallel distributed memory computers. In Proceedings of International Conference on Advances in Parallel and Distributed Computing (AFDC-97), March 1997. Shanghai, P. R. China.
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© 1997 Springer-Verlag Berlin Heidelberg
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Yang, T., Lin, HX. (1997). The highly parallel incomplete Gram-Schmidt preconditioner. In: Malyshkin, V. (eds) Parallel Computing Technologies. PaCT 1997. Lecture Notes in Computer Science, vol 1277. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63371-5_50
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DOI: https://doi.org/10.1007/3-540-63371-5_50
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