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

• Autorzy: Dorota Krawczyk-Stańdo , Marek Rudnicki
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2007
• Tom: 17
• Numer: 2
• Strony: 157-164

Daty

wydano

2007

otrzymano

2006-12-08

(nieznana)

2006-12-15

poprawiono

2007-04-14

Twórcy

autor

Dorota Krawczyk-Stańdo

• Center of Mathematics and Physics, Technical University of Łódź, ul. Al. Politechniki 11, 90–924 Łódź, Poland

autor

Marek Rudnicki

• Institute of Computer Science, Technical University of Łódź, ul. Wólczańska 215, 90–924 Łódź, Poland

Bibliografia

• 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.
• 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 Electry 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 electromagnetism (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