Abstract
For the solutions of linear systems of equations with unsymmetric coefficient matrices, we propose an improved version of the quasi-minimal residual (IQMR) method by using the Lanczos process as a major component combining elements of numerical stability and parallel algorithm design. For Lanczos process, stability is obtained by a coupled two-term procedure that generates Lanczos vectors normalized to unit length. The algorithm is derived in such a way that all inner products and matrix-vector multiplications of a single iteration step are independent, subsequently communication time required for inner products can be overlapped efficiently with computation time. Therefore, the cost of global communication on parallel distributed memory computers is significantly reduced. The resulting IQMR algorithm preserves the favorable properties of the Lanczos process without increasing computational costs. The efficiency of this method is demonstrated by numerical experimental results carried out on a massively parallel distributed memory computer, the Parsytec GC/PowerPlus.
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H. M. Bucker and M. Sauren. A parallel version of the quasi-minimal residual method based on coupled two-term recurrences. In Proceedings of Workshop on Applied Parallel Computing in Industrial Problems and Optimization (Para96). Technical University of Denmark, Lyngby, Denmark, Springer-Verlag, August 1996.
H. M. Bucker and M. Sauren. A parallel version of the unsymmetric Lanczos algorithm and its application to QMR. Technical Report KFA-ZAM-IB-9606, Central Institute for Applied Mathematics, Research Centre Julich, Germany, March 1996.
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.
J. J. Dongarra, I. S. Duff, D. C. Sorensen, and H. A. van der Vorst. Solving Linear Systems on Vector and Shared Memory Computers. SIAM, Philadelphia, PA, 1991.
R. W. Freund, G. H. Golub, and N. Nachtigal. Iterative solution of linear systems. Acta Numerica, pages 57–100, 1991.
R. W. Freund, M. H. Gutknecht, and N. M. Nachtigal. An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices. SIAM Journal on Scientific and Statistical Computing, 14:137–158, 1993.
R. W. Freund and N. M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik, 60:315–339, 1991.
R. W. Freund and N. M. Nachtigal. An implementation of the QMR method based on coupled two-term recurrences. SIAM Journal on Scientific and Statistical Computing, 15(2):313–337, 1994.
S. K. Kim and A. T. Chronopoulos. An efficient nonsymmetric Lanczos method on parallel vector computers. Journal of Computational and Applied Mathematics, pages 357–374, 1992.
C. Lanczos. An iteration method for the solution of the eigenvalues problem of linear differential and integral operators. Journal of Research of National Bureau of Standards, 45:255–282, 1950.
B. N. Parlett, D. R. Taylor, and Z. A. Liu. A look-ahead Lanczos algorithm for unsymmetric matrices. Mathematics of Computation, 44:105–124, 1985.
D. R. Taylor. Analysis of the look ahead Lanczos algorithm for unsymmetric matrices. PhD thesis, Department of Mathematics, University of California at Berkeley, November 1982.
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© 1997 Springer-Verlag Berlin Heidelberg
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Yang, TR., Lin, HX. (1997). The improved quasi-minimal residual method on massively distributed memory computers. In: Hertzberger, B., Sloot, P. (eds) High-Performance Computing and Networking. HPCN-Europe 1997. Lecture Notes in Computer Science, vol 1225. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0031611
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DOI: https://doi.org/10.1007/BFb0031611
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