Skip to main content
SpringerLink
Log in

A multilevel block incomplete cholesky preconditioner for solving normal equations in linear least squares problems

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. A. Ajiz and A. Jennings,A robust incomplete Choleski-conjugate gradient algorithm, Int. J. Numer. Methods Engrg.,20 (1984), 949–966.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Ashby,Minimax polynomial preconditioning for Hermitian linear systems, SIAM J. Matrix Anal. Appl.,12 (1991), 766–789.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. W. Berry and R. J. Plemmons, Algorithms and experiments for structural mechanics on high-performance architectures, Comput. Methods Appl. Mech. Engrg.,64 (1987), 487–507.

    Article  MATH  Google Scholar 

  4. A. Björck,Numerical Methods for Least-Squares Problems, SIAM, Philadelphia, PA, 1996.

    MATH  Google Scholar 

  5. A. Björck and J. Y. Yuan,Preconditioners for least squares problems by LU factorization, Elect. Trans. Numer. Anal.,8 (1999), 26–35.

    MATH  Google Scholar 

  6. I. Chio, C. L. Monna, and D. Shanno,Future development of a primal-dual interior point method, ORSA J. Comput.,2 (1990), 304–311.

    Google Scholar 

  7. M. Fortin and R. Glowinski,Augmented Lagrange Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland, Amsterdam, 1983.

    Google Scholar 

  8. R. Freund,A note on two SOR methods for sparse least squares problems, Linear Algebra Appl.,88/89 (1987), 211–221.

    Article  MathSciNet  Google Scholar 

  9. G. H. Golub and J. M. Ortega,Scientific Computing: An Introduction with Parallel Computing, Academic Press, Boston, MA, 1993.

    MATH  Google Scholar 

  10. G. H. Golub and C. F. van Loan,Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 3rd edition, 1996.

    MATH  Google Scholar 

  11. M. T. Heath, R. J. Plemmons, and R. C. Ward,Sparse orthogonal schemes for structural optimization using force method, SIAM J. Sci. Statist. Comput.,5 (1984), 514–532.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. R. Hestenes and E. Stiefel,Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards,49 (1952), 409–436.

    MATH  MathSciNet  Google Scholar 

  13. A. Jennings and M. A. Ajiz,Incomplete methods for solving A T Ax=b, SIAM J. Sci. Statist. Comput.,5 (1984), 978–987.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. T. Jones and P. E. Plassmann,An incomplete Cholesky factorization, ACM Trans. Math. Software,21 (1995), 5–17.

    Article  MATH  MathSciNet  Google Scholar 

  15. E. G. Kolata,Geodesy: dealing with an enormous computer task, Science,200 (1978), 421–422.

    Article  Google Scholar 

  16. C. Lin and J. J. Moré,Incomplete Cholesky factorization with limited memory, SIAM J. Sci. Comput.,21 (1999), 24–45.

    Article  MATH  MathSciNet  Google Scholar 

  17. T. A. Manteuffel,An incomplete factorization technique for positive definite linear systems, Math. Comput.,34 (1980), 473–497.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. A. Meijerink and H. A. van der Vorst,An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput.,31 (1977), 148–162.

    Article  MATH  Google Scholar 

  19. C. C. Paige and M. A. Saunders,LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software,8 (1982), 43–71.

    Article  MATH  MathSciNet  Google Scholar 

  20. J. R. Rice,PARVEC workshop on very large least squares problems and supercomputers, Technical Report CSD-TR 464, Department of Computer Science, Purdue University, West Lafayette, IN, 1983.

    Google Scholar 

  21. Y. Saad,Preconditioning techniques for nonsymmetric and indefinite linear systems, J. Comput. Appl. Math.,24 (1988), 89–105.

    Article  MATH  MathSciNet  Google Scholar 

  22. Y. Saad,ILUT: a dual threshold incomplete LU preconditioner, Numer. Linear Algebra Appl.,1 (1994), 387–402.

    Article  MATH  MathSciNet  Google Scholar 

  23. Y. Saad,Heratice Methods for Sparse Linear Systems, PWS Publishing, New York, NY, 1996.

    Google Scholar 

  24. Y. Saad and J. Zhang,BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems, SIAM J. Sci. Comput.,20 (1999), 2103–2121.

    Article  MATH  MathSciNet  Google Scholar 

  25. Y. Saad and J. Zhang,BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices, SIAM J. Matrix Anal. Appl.,21 (1999), 279–299.

    Article  MATH  MathSciNet  Google Scholar 

  26. C. Shen and J. Zhang,Parallel two level block ILU preconditioning techniques for solving large sparse linear systems, Paral. Comput.,28 (2002), 1451–1475.

    Article  MATH  MathSciNet  Google Scholar 

  27. G. W. Stewart,Matrix Algorithms, Volume I: Basic Decompositions, SIAM, Philadelphia, PA, 1998.

    Google Scholar 

  28. X. Wang,Incomplete Factorization Preconditioning for Linear Least Squares Problems, PhD thesis, University of Illinois, Urbana-Champaign, IL, 1993.

    Google Scholar 

  29. X. Wang, K. A. Gallivan, and R. Bramley,CIMGS: an incomplete orthogonal factorization preconditioner, SIAM J. Sci. Comput.,18 (1997), 516–536.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory for High Performance Scientific Computing and Computer Simulation, Department of Computer, University of Kentucky, 773 Anderson Hall, 40506-0046, Lexington, KY, USA

    Jun Zhang & Tong Xiao

Authors
  1. Jun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tong Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported in part by the U.S. National Science Foundation under grants CCR-9902022, CCR-9988165, and CCR-0092532.

Jun Zhang received a Ph.D. from the George Washington University in 1997. He is an Associate Professor of Computer Science and Director of the Laboratory for High Performance Scientific Computing and Computer Simulation at the University of Kentucky. His research interests include large scale parallel and scientific computing, numerical simulation, iterative and preconditioning techniques for large scale matrix computation. Dr. Zhang is associate editor and on the editorial boards of three international journals in computer simulation and computational mathematics, and is on the program committees, of a few international conferences. His research work is currently funded by the U.S. National Science Foundation, the U.S. Department of Energy Office of Science, Japanese Research Organization for Information Science & Technology (RIST), and the University of Kentucky Research Committee. He is recipient of the U.S. National Science Foundation CAREER Award and several other awards.

Tong Xiao received an M.S. degree in Computer Science from the University of Kentucky in 2000.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, J., Xiao, T. A multilevel block incomplete cholesky preconditioner for solving normal equations in linear least squares problems. JAMC 11, 59–80 (2003). https://doi.org/10.1007/BF02935723

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02935723

AMS subject classifications

Key words and phrases

Search

Navigation