Summary
This paper describes the calculation of the least square inverse of sparse matrices and the use of load balancing schemes for parallel processing in a heterogeneous environment. Due to the variation of number of non zero entries to be calculated row wise, as well as the difference of processor speeds, load imbalance may occur and have an impact on the performance. To improve the performance by keeping processors as busy as possible the redistribution of tasks and data is needed. We present an architecture and implementation outlines of a few load balancing schemes featured with one-sided communications in a framework of multithreading. We show through our tests that the use of load balancing schemes can improve the performance in some cases.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. N. Banerjee and M. W. Benson. “An approximate inverse based multigrid approach to biharmonic problem”. Inter. J. Computer Math., 40:201–210, 1991.
M. W. Benson. “Approximate inverses for solving large sparse linear systems”. Master’s thesis, Lakehead University, 1973.
M. W. Benson and P. O. Frederickson. “Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems”. Utilitas Mathematica, 22:127–140, 1982.
M. W. Benson and P. O. Frederickson. “Fast parallel algorithms for the Moore-Penrose pseudo inverse”. In M. T. Heath, editor, Hypercube Multiprocessors, pages 597–604. SIAM, 1987.
M. W. Benson, J. Krettmann, and M. Wright. “Parallel algorithms for the solution of certain large sparse linear systems”. Int. J. Comput. Math., 16:245–260, 1984.
A. Brandt. “Algebraic multigrid theory: Symmetric cases”. App. Math. Comp., 19:23–56, 1986.
B. Carpentieri, I. S. Duff, and L. Giraud. “Sparse pattern selection strategies for robust Frobenius norm minimization preconditioners in electromagnetism”. Technical Report TR/PA/00/05, CERFACS, 42 Avenue G. Coriolis, 31057 Toulouse Celex, France, 2000.
E. Chow. “parallel implementation and practical use of sparse approximate inverses with a priori sparsity patterns”. Int. J. High Perf. Comput. Appl., 15:56–74, 2001.
G. Cybenko. “Dynamic load balancing for distributed memory multiprocessors”. J. Parallel and Distributed Computing, 7:279–301, 1989.
R. Elsasser, B. Monien, and R. Preis. “Diffusive Load Balancing Schemes on Heterogeneous Networks”. In Proc. SPAA 2000, Maine, 2000.
P. O. Frederickson. “Fast approximate inverse of large sparse linear systems”. Technical Report Mathematics Report #7-75, Department of Mathematical Sciences, Lakehead University, Canada, 1975.
B. Ge. “On the patterns of the least squares inverses of sparse matrices”, 1992. Essay, Lakehead University, Canada.
M. Grote and T. Huckle. “Parallel preconditioning with sparse approximate inverses”. SIAM J. Sci. Comput., 18(3):838–853, 1997.
Z. Lan, V. E. Taylor, and G. Bryan. “Dynamic load balancing of SAMR applications on distributed systems”. In Proc. 2001 ACM/IEEE conference on Supercomputing, Denver, CO, 2001.
J. W. Ruge and K. Stüben. “Algebraic Multigrid”. In S. F. McCormic, editor, Multigrid Methods, volume 66. SIAM, Philadelphia, 1987.
K. Schloegel, G. Karypis, and V. Kumar. “Multilevel diffusion schemes for repartitioning of adaptive meshes”. J. Parallel and Distributed Computing, 47:109–124, 1997.
W-P. Tang. “Toward an effective sparse approximate inverse preconditioner”. SIAM J. Matrix Anal. Appl., 20(4):970–986, 1999.
R. V. van Nieuwpoort, T. Kielmann, and H. E. Bal. “Efficient load balancing for wide-area divide-and-conquer applications”. In Proceedings of the eighth ACM SIGPLAN symposium on Principles and practices of parallel programming, pages 34–43, Snowbird, Utah, 2001. ACM.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ge, B. (2005). Computing the Least Squares Inverse of Sparse Matrices on a Network of Clusters. In: Zhang, W., Tong, W., Chen, Z., Glowinski, R. (eds) Current Trends in High Performance Computing and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27912-1_4
Download citation
DOI: https://doi.org/10.1007/3-540-27912-1_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25785-1
Online ISBN: 978-3-540-27912-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)