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International Journal of Applied Mathematics and Computer Science

Journal of University of Zielona Gora and Lubuskie Scientific Society


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Regularization Parameter Selection in Discrete Ill-Posed Problems — The Use of the U-Curve

Dorota Krawczyk-Stańdo1 / Marek Rudnicki1

Center of Mathematics and Physics, Technical University of Łódź, ul. Al. Politechniki 11, 90-924 Łódź, Poland1

Institute of Computer Science, Technical University of Łódź, ul. Wólczańska 215, 90-924 Łódź, Poland2

This content is open access.

Citation Information: International Journal of Applied Mathematics and Computer Science. Volume 17, Issue 2, Pages 157–164, ISSN (Print) 1641-876X, DOI: 10.2478/v10006-007-0014-3, July 2007

Publication History

Published Online:
2007-07-17

Regularization Parameter Selection in Discrete Ill-Posed Problems — The Use of the U-Curve

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.

Keywords: ill-posed problems; Tikhonov regularization; regularization parameter; L-curve; U-curve

  • Groetsch N. (1984): The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind. — London: Pitman.

  • 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. [CrossRef]

  • 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.

  • Hansen P.C. (1993): Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems. — Report UNIC-92-03

  • 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.

  • Lawson C.L. and Hanson R.J. (1974): Solving Least Squares Problems. — Englewood Cliffs, NJ: Prentice-Hall.

  • Neittaanmaki P., Rudnicki M. and Savini A. (1996): Inverse Problems and Optimal Design in Electrity and Magnetism. — Oxford: Clarendon Press.

  • Regińska T. (1996): A regularization parameter in discrete ill-posed problems. — SIAM J. Sci. Comput., Vol. 17, No. 3, pp. 740-749.

  • 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 electro-magnetism (M. Rudnicki and S. Wiak, Ed.). — Dordrecht, Kluwer, pp. 113-124.

  • 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. [CrossRef]

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