Numerical Algorithms

, Volume 60, Issue 4, pp 683–696 | Cite as

On the reduction of Tikhonov minimization problems and the construction of regularization matrices

  • L. Dykes
  • L. Reichel
Original Paper


Tikhonov regularization replaces a linear discrete ill-posed problem by a penalized least-squares problem, whose solution is less sensitive to errors in the data and round-off errors introduced during the solution process. The penalty term is defined by a regularization matrix and a regularization parameter. The latter generally has to be determined during the solution process. This requires repeated solution of the penalized least-squares problem. It is therefore attractive to transform the least-squares problem to simpler form before solution. The present paper describes a transformation of the penalized least-squares problem to simpler form that is faster to compute than available transformations in the situation when the regularization matrix has linearly dependent columns and no exploitable structure. Properties of this kind of regularization matrices are discussed and their performance is illustrated.


Ill-posed problem Tikhonov regularization Regularization matrix GSVD 


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  1. 1.
    Bai, Z.: The CSD, GSVD, Their Applications and Computation. IMA preprint 958, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN (1992)Google Scholar
  2. 2.
    Bai, Z., Demmel, J.W.: Computing the generalized singular value decomposition. SIAM J. Sci. Comput. 14, 1464–1486 (1993)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Extrapolation techniques for ill-conditioned linear systems. Numer. Math. 81, 1–29 (1998)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear systems. Numer. Math. 94, 203–228 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for linear systems with application to regularization. Numer. Algorithms 49, 85–104 (2008)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for the regularization of least squares problems. Numer. Algorithms 51, 61–76 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Calvetti, D., Reichel, L., Shuibi, A.: Invertible smoothing preconditioners for linear discrete ill-posed problems. Appl. Numer. Math. 54, 135–149 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. Dover, Mineola (1991)MATHGoogle Scholar
  9. 9.
    Eldén, L.: Algorithms for the regularization of ill-conditioned least squares problems. BIT 17, 134–145 (1977)MATHCrossRefGoogle Scholar
  10. 10.
    Eldén, L.: A weighted pseudoinverse, generalized singular values, and constrained least squares problems. BIT 22, 487–501 (1982)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)MATHCrossRefGoogle Scholar
  12. 12.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  13. 13.
    Hansen, P.C.: Regularization, GSVD and truncated GSVD. BIT 29, 491–504 (1989)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hansen, P.C.: Regularization tools version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189–194 (2007)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)CrossRefGoogle Scholar
  16. 16.
    Morigi, S., Reichel, L., Sgallari, F.: Orthogonal projection regularization operators. Numer. Algorithms 44, 99–114 (2007)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. ACM 9, 84–97 (1962)MATHCrossRefGoogle Scholar
  18. 18.
    Reichel, L., Ye, Q.: Simple square smoothing regularization operators. Electron. Trans. Numer. Anal. 33, 63–83 (2009)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA

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