Skip to main content
SpringerLink
Log in

Lanczos based preconditioner for discrete ill-posed problems

  • Published:
Computing Aims and scope Submit manuscript

Abstract

In this paper we use the Lanczos process for preconditioning discrete ill-posed problems. We show that by few steps of this process one can obtain a well qualified and efficient preconditioner. This is a general method in the sense that it is not limited only to special structured matrices and the matrix–vector multiplications can be carried out in O(n) operations. Also even in problems with structured matrices this preconditioner performs more efficiently than the circulant and Kronecker product approximate 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. Bjorck A (1996) Numerical methods for least squares problems. SIAM, Philadelphia

    Google Scholar 

  2. Calvetti D, Golub GH, Reichel L (1999) Estimation of the L-curve via Lanczos bidiagonalization. BIT 39: 603–619

    Article  MATH  MathSciNet  Google Scholar 

  3. Calvetti D, Reichel L, Shuibi A (2004) L-curve and curvature bounds for Tikhonov regularization. Numer Algorithms 35(2–4): 301–314

    Article  MATH  MathSciNet  Google Scholar 

  4. Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer, Dordrecht

    MATH  Google Scholar 

  5. Golub GH, Luk FT, Overton ML (1981) A block Lanczos method for computing the singular values and corresponding singular vectors of a matrix. AMC Trans Math Softw 7(2): 149–169

    Article  MATH  MathSciNet  Google Scholar 

  6. Golub GH, Reinsch C (1970) Singular value decomposition and least squares solutions. Numer Math 14(5): 403–420

    Article  MATH  MathSciNet  Google Scholar 

  7. Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  8. Hanke M (1995) Conjugate gradient type methods for ill-posed problems. In: Pitman research notes in mathematics, no 327. Longman, Harlow

  9. Hanke M, Hansen PC (1993) Regularization methods for large-scale problems. Surv Math Ind 3(4): 253–315

    MATH  MathSciNet  Google Scholar 

  10. Hanke M, Nagy J (1998) Inverse Toeplitz preconditioners for ill-posed problems. Linear Algebra Appl 284: 137–156

    Article  MATH  MathSciNet  Google Scholar 

  11. Hanke M, Nagy J, Plemmons R (1993) Preconditioned iterative regularization methods for ill-posed problems. In: Reichel L, Ruttan A, Varga RS (eds) Numerical linear algebra. de Gruyter, Berlin, pp 141–163

    Google Scholar 

  12. Hansen PC (1994) Regularization tools: a MATLAB package for analysis and solution of discrete ill-posed problems. Numer Algorithms 6:1–35. Software is available in Netlib at http://www.netlib.org

  13. Hansen PC (1998) Rank-deficient and discrete ill-posed problems. SIAM, Philadelphia

    Google Scholar 

  14. Hansen PC, Jensen TK, Rodriguez G (2007) An adaptive pruning algorithm for the discrete L-curve criterion. J Comput Appl Math 198: 483–492

    Article  MATH  MathSciNet  Google Scholar 

  15. Kamm J, Nagy JG (1998) Kronecker product and SVD approximations in image restoration. Linear Algebra Appl 284: 177–192

    Article  MATH  MathSciNet  Google Scholar 

  16. Kamm J, Nagy JG (2000) Optimal Kronecker product approximation of block Toeplitz matrices. SIAM J Matrix Anal Appl 22: 155–172

    Article  MATH  MathSciNet  Google Scholar 

  17. Kilmer ME, Oleary DP (2001) Choosing regularization parameter in iterative methods for ill-posed problems. SIAM J Matrix Anal Appl 22: 1204–1221

    Article  MATH  MathSciNet  Google Scholar 

  18. Nagy JG, Kilmer ME (2006) Kronecker product approximation for preconditioning in three-dimensional imaging applications. IEEE Trans Image Process 15: 604–614

    Article  MathSciNet  Google Scholar 

  19. Paige CC (1974) Bidiagonalization of matrices and solution of linear equations. SIAM J Numer Anal 11: 197–209

    Article  MATH  MathSciNet  Google Scholar 

  20. RestoreTools (2002) An object oriented Matlab package for image restoration. http://www.mathcs.emory.edu/nagy/RestoreTools

  21. Saad Y (2004) Iterative methods for sparse linear systems, 2nd edn. SIAM, Philadelphia

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-175, Tehran, Iran

    Mansoor Rezghi & S. M. Hosseini

Authors
  1. Mansoor Rezghi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. M. Hosseini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. M. Hosseini.

Additional information

Communicated by X. Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rezghi, M., Hosseini, S.M. Lanczos based preconditioner for discrete ill-posed problems. Computing 88, 79–96 (2010). https://doi.org/10.1007/s00607-010-0090-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-010-0090-3

Keywords

Search

Navigation