Numerical Algorithms

, Volume 35, Issue 2–4, pp 301–314 | Cite as

L-Curve and Curvature Bounds for Tikhonov Regularization

  • D. Calvetti
  • L. Reichel
  • A. Shuibi


The L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems by Tikhonov regularization. However, the computational effort required to determine the L-curve and its curvature can be prohibitive for large-scale problems. Recently, inexpensively computable approximations of the L-curve and its curvature, referred to as the L-ribbon and the curvature-ribbon, respectively, were proposed for the case when the regularization operator is the identity matrix. This note discusses the computation and performance of the L- and curvature-ribbons when the regularization operator is an invertible matrix.

ill-posed problem regularization L-curve 


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  1. [1]
    M.L. Baart, The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned leastsquares problems, IMA J. Numer. Anal. 2 (1982) 241–247.Google Scholar
  2. [2]
    Å. Björck and L. Eldén, Methods in numerical linear algebra for ill-posed problems, Report, Department of Mathematics, Linköping University, Linköping, Sweden (1979).Google Scholar
  3. [3]
    D. Calvetti, G.H. Golub and L. Reichel, Estimation of the L-curve via Lanczos bidiagonalization, BIT 39 (1999) 603–619.Google Scholar
  4. [4]
    D. Calvetti, P.C. Hansen and L. Reichel, L-curve curvature bounds via Lanczos bidiagonalization, Elec. Trans. Numer. Anal. 14 (2002) 20–35. Available at the Web site Scholar
  5. [5]
    D. Calvetti, S. Morigi, L. Reichel and F. Sgallari, Tikhonov regularization and the L-curve for large, discrete ill-posed problems, J. Comput. Appl. Math. 123 (2000) 423–446.Google Scholar
  6. [6]
    D. Calvetti, S. Morigi, L. Reichel and F. Sgallari, An L-ribbon for large underdetermined linear discrete ill-posed problems, Numer. Algorithms 25 (2000) 89–107.Google Scholar
  7. [7]
    L.M. Delves and J.L. Mohamed, Computational Methods for Integral Equations (Cambridge Univ. Press, Cambridge, 1985).Google Scholar
  8. [8]
    L. Eldén, Algorithms for the regularization of ill-conditioned least squares problems, BIT 17 (1977) 134–145.Google Scholar
  9. [9]
    P.C. Hansen, Regularization tools: A Matlab package for analysis and solution of discrete illposed problems, Numer. Algorithms 6 (1994) 1–35. Software is available in Netlib at the Web site Scholar
  10. [10]
    P.C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (SIAM, Philadelphia, PA, 1998).Google Scholar
  11. [11]
    P.C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, in: Computational Inverse Problems in Electrocardiology, ed. P. Johnston, Advances in Computational Bioengineering, Vol. 4 (WIT Press, Southampton, 2000) pp. 119–142.Google Scholar
  12. [12]
    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.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • D. Calvetti
    • 1
  • L. Reichel
    • 2
  • A. Shuibi
    • 2
  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA
  2. 2.Department of Mathematical SciencesKent State UniversityKentUSA

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