Abstract
We derive a new parallel communication-avoiding matrix powers algorithm for matrices of the form \(A=D+USV^H\), where \(D\) is sparse and \(USV^H\) has low rank and is possibly dense. We demonstrate that, with respect to the cost of computing \(k\) sparse matrix-vector multiplications, our algorithm asymptotically reduces the parallel latency by a factor of \(O(k)\) for small additional bandwidth and computation costs. Using problems from real-world applications, our performance model predicts up to \(13\times \) speedups on petascale machines.
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
References
Bebendorf, M.: A means to efficiently solve elliptic boundary value problems. In: Bart, T., Griebel, M., Keyes, D., Nieminen, R., Roose, D., Schlick, T. (eds.) Hierarchical Matrices. LNCS, vol. 63, pp. 49–98. Springer, Heidelberg (2008)
Chan, E., Heimlich, M., Purkayastha, A., Van De Geijn, R.: Collective communication: theory, practice, and experience. Concurrency Comput.: Pract. Exper. 19, 1749–1783 (2007)
Chandrasekaran, S., Dewilde, P., Gu, M., Lyons, W., Pals, T.: A fast solver for HSS representations via sparse matrices. SIAM J. Matrix Anal. Appl. 29, 67–81 (2006)
Demmel, J., Hoemmen, M., Mohiyuddin, M., Yelick, K.: Avoiding communication in computing Krylov subspaces. Technical report UCB/EECS-2007-123, University of California-Berkeley (2007)
Hoemmen, M.: Communication-avoiding Krylov subspace methods. Ph.D. thesis, University of California-Berkeley (2010)
Hong, J., Kung, H.: I/O complexity: the red-blue pebble game. In: Proceedings of the 13th ACM Symposium on Theory of Computing, pp. 326–333. ACM, New York (1981)
Knight, N., Carson, E., Demmel, J.: Exploiting data sparsity in parallel matrix powers computations. Technical report UCB/EECS-2013-47, University of California-Berkeley (2013)
Kriemann, R.: Parallele Algorithmen für \(\cal H\)-Matrizen. Ph.D. thesis, Christian-Albrechts-Universität zu Kiel (2005)
Leiserson, C., Rao, S., Toledo, S.: Efficient out-of-core algorithms for linear relaxation using blocking covers. J. Comput. Syst. Sci. Int. 54, 332–344 (1997)
Mohiyuddin, M.: Tuning hardware and software for multiprocessors. Ph.D. thesis, University of California-Berkeley (2012)
Mohiyuddin, M., Hoemmen, M., Demmel, J., Yelick, K.: Minimizing communication in sparse matrix solvers. In: Proceedings of the Conference on High Performance Computing Networking, Storage, and Analysis, pp. 36:1–36:12. ACM, New York (2009)
Philippe, B., Reichel, L.: On the generation of Krylov subspace bases. Appl. Numer. Math. 62, 1171–1186 (2012)
Wang, S., Li, X., Xia, J., Situ, Y., de Hoop, M.: Efficient scalable algorithms for hierarchically semiseparable matrices. SIAM J. Sci. Comput. (2012, under review)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Knight, N., Carson, E., Demmel, J. (2014). Exploiting Data Sparsity in Parallel Matrix Powers Computations. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2013. Lecture Notes in Computer Science(), vol 8384. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55224-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-55224-3_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-55223-6
Online ISBN: 978-3-642-55224-3
eBook Packages: Computer ScienceComputer Science (R0)