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

• Autorzy: Krawczyk-Stańdo D. , Rudnicki M.
• 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 \alfa 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 \alfa, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.

Słowa kluczowe

EN

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

PL

problem niewłaściwie postawiony; regularyzacja Tichonowa; parametr regularyzacji

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2007
• Tom: Vol. 17, no 2
• Strony: 157--164
• Opis fizyczny: Bibliogr. 10 poz., wykr.

Twórcy

autor

Krawczyk-Stańdo D.

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

autor

Rudnicki M.

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

Bibliografia

• [1] Groetsch N. (1984): The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind.-London: Pitman.
• [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.
• [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.
• [7] Neittaanmaki P., Rudnicki M. and Savini A. (1996): Inverse Problems and Optimal Design in Electrity and Magnetism. - Oxford: Clarendon Press.
• [8] Regińska T. (1996): A regularization parameter in discrete illposed problems. - SIAM J. Sci. Comput., Vol. 17, No. 3, pp. 740-749.
• [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.