Abstract
We study the determination of large and sparse derivative matrices using row and column compression. This sparse matrix determination problem has rich combinatorial structure which must be exploited to effectively solve any reasonably sized problem. We present a new algorithm for computing a two-sided compression of a sparse matrix. We give new lower bounds on the number of matrix-vector products needed to determine the matrix. The effectiveness of our algorithm is demonstrated by numerical testing on a set of practical test instances drawn from the literature.
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
Notes
- 1.
A row compression is a one-sided compression where only seed matrix S is defined.
References
Coleman, T.F., Moré, J.J.: Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal. 20 (1), 187–209 (1983)
Coleman, T.F., Verma, A.: The efficient computation of sparse Jacobian matrices using automatic differentiation. SIAM J. Sci. Comput. 19 (4), 1210–1233 (1998)
Curtis, A.R., Powell, M.J.D., Reid, J.K.: On the estimation of sparse Jacobian matrices. IMA J. Appl. Math. 13 (1), 117–119 (1974)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Griewank, A., Toint, Ph.L.: On the unconstrained optimization of partially separable objective functions. In: Powell, M.J.D. (ed.), Nonlinear Optimization, pp. 301–312. Academic, London (1982)
Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2008)
Hossain, A.K.M.S.: On the computation of sparse Jacobian matrices and newton steps. Ph.D. Dissertation, Department of Informatics, University of Bergen (1998)
Hossain, A.K.M.S., Steihaug, T.: Computing a sparse Jacobian matrix by rows and columns. Optim. Methods Softw. 10, 33–48 (1998)
Hossain, S., Steihaug, T.: Graph models and their efficient implementation for sparse Jacobian matrix determination. Discret. Appl. Math. 161 (2), 1747–1754 (2013)
Juedes, D., Jones, J.: Coloring Jacobians revisited: a new algorithm for star and acyclic bicoloring. Optim. Methods Softw. 27, 295–309 (2012)
Leighton, F.T.: A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 84, 489–505 (1979)
Acknowledgements
This research is supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (Individual).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Gaur, D.R., Hossain, S., Saha, A. (2016). Determining Sparse Jacobian Matrices Using Two-Sided Compression: An Algorithm and Lower Bounds. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R. (eds) Mathematical and Computational Approaches in Advancing Modern Science and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30379-6_39
Download citation
DOI: https://doi.org/10.1007/978-3-319-30379-6_39
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30377-2
Online ISBN: 978-3-319-30379-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)