Advertisement

Numerical Aspects in Locating the Corner of the L-curve

  • Valia Guerra
  • Victoria Hernández

Abstract

The L-curve method can be used to choose the regularization parameter, when discrete ill-posed problems are solved by a regularization method. In this paper, we propose a novel numerical algorithm for locating the corner of the L-curve when only a finite set of its points is known. The proposed technique is based on conic section fitting. The corner point is chosen to be the nearest one to the shoulder point of the conic. The performance of the method is assessed by comparing its results with those obtained with another approach reported previously by Hansen [6].

Keywords

Singular Value Decomposition Regularization Method Corner Point Tikhonov Regularization Conic Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Farin G. (1988), Curves and Surfaces for Computer Aided Geometric Design, Academic Press, New YorkGoogle Scholar
  2. 2.
    Guerra V., Hernández V. (1999), Numerical Issues in locating the corner of the L-curve, Technical Report 99–57, Center of Mathematics and Theoretical Physics, HavanaGoogle Scholar
  3. 3.
    Grefer H.(1999), An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Math. Comput. 49:.507–522Google Scholar
  4. 4.
    Hansen P.C. (1987), The truncated SVD as a method for regularization, BIT, 27: 354–553CrossRefGoogle Scholar
  5. 5.
    Hansen P.C. (1992), Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34: 561–580CrossRefGoogle Scholar
  6. 6.
    Hansen P. C., O’Leary D.P. (1993), The use of the L-curve in the Regularization of discrete ill-posed problems, SIAM J. Sci. Comput. Vol 14, No. 6: 1487–1503CrossRefGoogle Scholar
  7. 7.
    Hansen P. C. (1998), Regularization Tools. A MATLAB package for Analysis and Solution of Discrete Ill-posed Problems, Technical Report, Dept. of Mathematical Modelling, Technical University of DenmarkGoogle Scholar
  8. 8.
    Hansen P.C. (1995), Test matrices for regularization methods, SIAM J. Sci. Comput., 16: 506–512CrossRefGoogle Scholar
  9. 9.
    Hansen P.C. (1998), Rank Deficient and Discrete Ill-posed problems, SIAMCrossRefGoogle Scholar
  10. 10.
    Hernandez V., Estrada J., Barrera P. (1998), On the Euclidean distance from a point to a conic. Revista Integración, Vol 15, No.1: 45–61Google Scholar
  11. 11.
    Morozov V. A. (1966), On the solution of functional equations by the method of regularization. Soviet Math. Dokl, 7: 414–41Google Scholar
  12. 12.
    Neumaier A. (1998), Solving ill-conditioned and singular linear systems: A tutorial on Regularization, SIAM Rev. Vol. 40, No. 3: 636–666sCrossRefGoogle Scholar
  13. 13.
    Taubin G.(1994), Distance approximations for rasterizing implicit curves. ACM Trans on Graphics, Vol. 13, No. 1CrossRefGoogle Scholar
  14. 14.
    Tikhonov A.T., Arsenin V.Y. (1977), Solutions of Ill-Posed Problems, John Wiley, New YorkGoogle Scholar
  15. 15.
    Wahba G. (1977), Practical approximate solutions to linear operator equations when the data are noisy. SIAM J. Numer. Anal. 14Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Valia Guerra
    • 1
  • Victoria Hernández
    • 1
  1. 1.Centro de Matemática y Física TeóricaMinisterio de CienciaHavanaCuba

Personalised recommendations