Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 14, 2016

Preconditioning Techniques Based on the Birkhoff–von Neumann Decomposition

Michele Benzi EMAIL logo and Bora Uçar

Abstract

We introduce a class of preconditioners for general sparse matrices based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices. These preconditioners are aimed primarily at solving challenging linear systems with highly unstructured and indefinite coefficient matrices. We present some theoretical results and numerical experiments on linear systems from a variety of applications.

Award Identifier / Grant number: DMS-1418889

Award Identifier / Grant number: SOLHAR (ANR-13-MONU-0007)

Funding statement: The work of Michele Benzi was supported in part by NSF grant DMS-1418889. Bora Uçar was supported in part by French National Research Agency (ANR) project SOLHAR (ANR-13-MONU-0007).

Acknowledgements

We thank Alex Pothen for his contributions to this work. This work resulted from the collaborative environment offered by the Dagstuhl Seminar 14461 on High-Performance Graph Algorithms and Applications in Computational Science (November 9–14, 2014).

References

[1] Amestoy P. R., Duff I. S., Ruiz D. and Uçar B., A parallel matrix scaling algorithm, High Performance Computing for Computational Science – VECPAR 2008, Lecture Notes in Comput. Sci. 5336, Springer, Berlin (2008), 301–313. 10.1007/978-3-540-92859-1_27Search in Google Scholar

[2] Anzt H., Chow E. and Dongarra J., Iterative sparse triangular solves for preconditioning, Euro-Par 2015: Parallel Processing, Lecture Notes in Comput. Sci. 9233, Springer, Berlin (2015), 650–651. 10.1007/978-3-662-48096-0_50Search in Google Scholar

[3] Benzi M., Haws J. C. and Tuma M., Preconditioning highly indefinite and nonsymmetric matrices, SIAM J. Sci. Comput. 22 (2000), no. 4, 1333–1353. 10.1137/S1064827599361308Search in Google Scholar

[4] Birkhoff G., Tres observaciones sobre el algebra lineal, Univ. Nac. Tucumán Rev. Ser. A 5 (1946), 147–150. Search in Google Scholar

[5] Brualdi R. A., Notes on the Birkhoff algorithm for doubly stochastic matrices, Canad.Math. Bull. 25 (1982), no. 2, 191–199. 10.4153/CMB-1982-026-3Search in Google Scholar

[6] Brualdi R. A. and Gibson P. M., Convex polyhedra of doubly stochastic matrices. I: Applications of the permanent function, J. Combin. Theory Ser. A 22 (1977), no. 2, 194–230. 10.1016/0097-3165(77)90051-6Search in Google Scholar

[7] Brualdi R. A. and Ryser H. J., Combinatorial Matrix Theory, Encyclopedia Math. Appl. 39, Cambridge University Press, Cambridge, 1991. 10.1017/CBO9781107325708Search in Google Scholar

[8] Burkard R., Dell’Amico M. and Martello S., Assignment Problems, SIAM, Philadelphia, 2009. 10.1137/1.9780898717754Search in Google Scholar

[9] Chow E. and Patel A., Fine-grained parallel incomplete LU factorization, SIAM J. Sci. Comput. 37 (2015), no. 2, C169–C193. 10.1137/140968896Search in Google Scholar

[10] Davis T. A. and Hu Y., The University of Florida sparse matrix collection, ACM Trans. Math. Software 38 (2011), 10.1145/2049662.2049663. 10.1145/2049662.2049663Search in Google Scholar

[11] Dolan E. D. and Moré J. J., Benchmarking optimization software with performance profiles, Math. Program. 91 (2002), no. 2, 201–213. 10.1007/s101070100263Search in Google Scholar

[12] Duff I. S., Erisman A. M. and Reid J. K., Direct Methods for Sparse Matrices, 2nd ed., Oxford University Press, Oxford, 2017. 10.1093/acprof:oso/9780198508380.001.0001Search in Google Scholar

[13] Duff I. S. and Koster J., The design and use of algorithms for permuting large entries to the diagonal of sparse matrices, SIAM J. Matrix Anal. Appl. 20 (1999), no. 4, 889–901. 10.1137/S0895479897317661Search in Google Scholar

[14] Duff I. S. and Koster J., On algorithms for permuting large entries to the diagonal of a sparse matrix, SIAM J. Matrix Anal. Appl. 22 (2001), 973–996. 10.1137/S0895479899358443Search in Google Scholar

[15] Dufossé F. and Uçar B., Notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices, Linear Algebra Appl. 497 (2016), 108–115. 10.1016/j.laa.2016.02.023Search in Google Scholar

[16] Gabow H. N. and Tarjan R. E., Algorithms for two bottleneck optimization problems, J. Algorithms 9 (1988), no. 3, 411–417. 10.1016/0196-6774(88)90031-4Search in Google Scholar

[17] Gantmacher F. R., The Theory of Matrices. Vol. 2, Chelsea Publishing, New York, 1959. Search in Google Scholar

[18] Halappanavar M., Pothen A., Azad A., Manne F., Langguth J. and Khan A. M., Codesign lessons learned from implementing graph matching on multithreaded architectures, IEEE Computer 48 (2015), no. 8, 46–55. 10.1109/MC.2015.215Search in Google Scholar

[19] Horn R. A. and Johnson C. R., Matrix Analysis, 2nd ed., Cambridge University, Cambridge, 2013. Search in Google Scholar

[20] Knight P. A. and Ruiz D., A fast algorithm for matrix balancing, IMA J. Numer. Anal. 33 (2013), no. 3, 1029–1047. 10.1093/imanum/drs019Search in Google Scholar

[21] Knight P. A., Ruiz D. and Uçar B., A symmetry preserving algorithm for matrix scaling, SIAM J. Matrix Anal. Appl. 35 (2014), no. 3, 931–955. 10.1137/110825753Search in Google Scholar

[22] Manguoglu M., Koyutürk M., Sameh A. H. and Grama A., Weighted matrix ordering and parallel banded preconditioners for iterative linear system solvers, SIAM J. Sci. Comput. 32 (2010), no. 3, 1201–1216. 10.1137/080713409Search in Google Scholar

[23] Saad Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput. 14 (1993), no. 2, 461–469. 10.1137/0914028Search in Google Scholar

[24] Sinkhorn R. and Knopp P., Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math. 21 (1967), 343–348. 10.2140/pjm.1967.21.343Search in Google Scholar

[25] Varga R. S., Matrix Iterative Analysis, 2nd ed., Springer, Berlin, 2000. 10.1007/978-3-642-05156-2Search in Google Scholar

Received: 2016-10-26
Accepted: 2016-11-14
Published Online: 2016-12-14
Published in Print: 2017-4-1

© 2017 by De Gruyter

Downloaded on 9.12.2022 from https://www.degruyter.com/document/doi/10.1515/cmam-2016-0040/html
Scroll Up Arrow