Regularization parameter selection in discrete ill-posed problems - the use of the U-curve

Dorota Krawczyk-Stańdo; Marek Rudnicki

International Journal of Applied Mathematics and Computer Science (2007)

  • Volume: 17, Issue: 2, page 157-164
  • ISSN: 1641-876X

Abstract

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To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter α we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter α, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.

How to cite

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Krawczyk-Stańdo, Dorota, and Rudnicki, Marek. "Regularization parameter selection in discrete ill-posed problems - the use of the U-curve." International Journal of Applied Mathematics and Computer Science 17.2 (2007): 157-164. <http://eudml.org/doc/207827>.

@article{Krawczyk2007,
abstract = {To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter α we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter α, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.},
author = {Krawczyk-Stańdo, Dorota, Rudnicki, Marek},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {U-curve; regularization parameter; L-curve; Tikhonov regularization; ill-posed problems; -curve},
language = {eng},
number = {2},
pages = {157-164},
title = {Regularization parameter selection in discrete ill-posed problems - the use of the U-curve},
url = {http://eudml.org/doc/207827},
volume = {17},
year = {2007},
}

TY - JOUR
AU - Krawczyk-Stańdo, Dorota
AU - Rudnicki, Marek
TI - Regularization parameter selection in discrete ill-posed problems - the use of the U-curve
JO - International Journal of Applied Mathematics and Computer Science
PY - 2007
VL - 17
IS - 2
SP - 157
EP - 164
AB - To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter α we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter α, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.
LA - eng
KW - U-curve; regularization parameter; L-curve; Tikhonov regularization; ill-posed problems; -curve
UR - http://eudml.org/doc/207827
ER -

References

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  1. Groetsch N. (1984): The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind. - London: Pitman. Zbl0545.65034
  2. Hansen P.C. (1992): Analysis of discrete ill-posed problems by means of the L-curve.- SIAM Rev., Vol.34, No.4, pp.561-580. Zbl0770.65026
  3. Hansen P.C. and 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, pp.487-1503. 
  4. Hansen P.C. (1993): Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems. - Report UNIC-92-03 
  5. Krawczyk-Stańdo D. and Rudnicki M. (2005): Regularized synthesis of the magnetic field using the L-curve approach. - Int. J. Appl. Electromagnet. Mech., Vol.22, No.3-4, pp.233-242. 
  6. Lawson C.L. and Hanson R.J. (1974): Solving Least Squares Problems.- Englewood Cliffs, NJ: Prentice-Hall. Zbl0860.65028
  7. Neittaanmaki P., Rudnicki M. and Savini A. (1996): Inverse Problems and Optimal Design in Electry and Magnetism. - Oxford: Clarendon Press. Zbl0865.65090
  8. Regińska T. (1996): A regularization parameter in discrete ill-posed problems. - SIAM J. Sci. Comput., Vol.17, No.3, pp.740-749. Zbl0865.65023
  9. Stańdo J., Korotow S., Rudnicki M., Krawczyk-Stańdo D. (2003): The use of quasi-red and quasi-yellow nonobtuse refinements in the solution of 2-D electromagnetic, PDE's, In: Optimization and inverse problems in electromagnetism (M. Rudnicki and S. Wiak, Ed.). - Dordrecht,Kluwer, pp.113-124. 
  10. Wahba G. (1977): Practical approximate solutions to linear operator equations when data are noisy.- SIAM J. Numer. Anal., Vol.14, No.4, pp.651-667 Zbl0402.65032

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