Skip to main content
SpringerLink
Log in

A Golub–Kahan-Type Reduction Method for Matrix Pairs

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We describe a novel method for reducing a pair of large matrices \(\{A,B\}\) to a pair of small matrices \(\{H,K\}\). The method is an extension of Golub–Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when B is the identity matrix. Applications to Tikhonov regularization of large linear discrete ill-posed problems are described. In these problems the matrix A represents a discretization of a compact integral operator and B is a regularization matrix.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Arioli, M., Orban, D.: Iterative methods for symmetric quasi-definite linear systems. Part I: theory, Cahier du GERAD G-2013-32, GERAD (2013)

  2. Belge, M., Kilmer, M.E., Miller, E.L.: Efficient selection of multiple regularization parameters in a generalized L-curve framework. Inverse Probl. 28, 1161–1183 (2002)

    Article  Google Scholar 

  3. Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear systems. Numer. Math. 94, 203–224 (2003)

    Article  MATH  Google Scholar 

  4. Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for the regularization of least squares problems. Numer. Algorithms 51, 61–76 (2009)

    Article  MATH  Google Scholar 

  5. Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the L-curve for large discrete ill-posed problems. J. Comput. Appl. Math. 123, 423–446 (2000)

    Article  MATH  Google Scholar 

  6. Calvetti, D., Reichel, L.: Tikhonov regularization of large linear problems. BIT 43, 263–283 (2003)

    Article  MATH  Google Scholar 

  7. Eldén, L.: A weighted pseudoinverse, generalized singular values, and constrained least squares problems. BIT 22, 487–501 (1982)

    Article  MATH  Google Scholar 

  8. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  9. Gazzola, S., Novati, P.: Multi-parameter Arnoldi–Tikhonov methods. Electron. Trans. Numer. Anal. 40, 452–475 (2013)

    MATH  Google Scholar 

  10. Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston (1984)

    MATH  Google Scholar 

  11. Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  12. Hansen, P.C.: Regularization tools version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189–194 (2007)

    Article  MATH  Google Scholar 

  13. Hochstenbach, M.E., Reichel, L.: An iterative method for Tikhonov regularization with a general linear regularization operator. J. Integral Equ. Appl. 22, 465–482 (2010)

    Article  MATH  Google Scholar 

  14. Hoffnung, L., Li, R.-C., Ye, Q.: Krylov type subspace methods for matrix polynomials. Linear Algebra Appl. 415, 52–81 (2006)

    Article  MATH  Google Scholar 

  15. Kilmer, M.E., Hansen, P.C., Español, M.I.: A projection-based approach to general-form Tikhonov regularization. SIAM J. Sci. Comput. 29, 315–330 (2007)

    Article  MATH  Google Scholar 

  16. Kindermann, S.: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron. Trans. Numer. Anal. 38, 233–257 (2011)

    MATH  Google Scholar 

  17. Li, R.-C., Ye, Q.: A Krylov subspace method for quadratic matrix polynomials with application to constrained least squares problems. SIAM J. Matrix Anal. Appl. 25, 405–428 (2003)

    Article  MATH  Google Scholar 

  18. Lu, S., Pereverzev, S.V.: Multi-parameter regularization and its numerical realization. Numer. Math. 118, 1–31 (2011)

    Article  MATH  Google Scholar 

  19. Morigi, S., Reichel, L., Sgallari, F.: Orthogonal projection regularization operators. Numer. Algorithms 44, 99–114 (2007)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

    Article  Google Scholar 

  22. Phillips, D.L.: A Technique for the numerical solution of certain integral equations of the first kind. J. ACM 9, 84–97 (1962)

    Article  MATH  Google Scholar 

  23. Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Numer. Algorithms 63, 65–87 (2013)

    Article  MATH  Google Scholar 

  24. Reichel, L., Sgallari, F., Ye, Q.: Tikhonov regularization based on generalized Krylov subspace methods. Appl. Numer. Math. 62, 1215–1228 (2012)

    Article  MATH  Google Scholar 

  25. Reichel, L., Yu, X.: Tikhonov regularization via flexible Arnoldi reduction. BIT Numer. Math. doi:10.1007/s10543-014-0542-9

Download references

Acknowledgments

We would like to thank Kazim Khan for discussions and the referees for suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, TU Eindhoven, PO Box 513, 5600 MB, Eindhoven, The Netherlands

    Michiel E. Hochstenbach

  2. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    Lothar Reichel & Xuebo Yu

Authors
  1. Michiel E. Hochstenbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lothar Reichel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuebo Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lothar Reichel.

Additional information

Michiel E. Hochstenbach: Supported by an NWO Vidi research Grant.

Lothar Reichel and Xuebo Yu: Research supported in part by NSF Grant DMS-1115385.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hochstenbach, M.E., Reichel, L. & Yu, X. A Golub–Kahan-Type Reduction Method for Matrix Pairs. J Sci Comput 65, 767–789 (2015). https://doi.org/10.1007/s10915-015-9990-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-9990-x

Keywords

Search

Navigation