Abstract
Regularization methods are used for computing stable solutions to the ill-posed problems. The well-known form of regularization is that of Tikhonov in which the regularized solution is searched as a minimize of the weighted combination of the residual norm and a side constraint-controlled by the regularization parameter. For the practical choice of regularization parameter α we can use the well-known L-curve criterion, or introduced by us U-curve criterion. The efficiency of the approach is demonstrated on examples of synthesis of magnetic field. The paper is finished with the comparison of the two mentioned above methods made on numerical examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Wahba, Practical approximate solutions to linear operator equations when data are noisy, SIAM Journal of Numerical Analysis, Vol. 14, No. 4, pp. 651–667, 1977
N. Groetsch, The theory of Tikhonov regularization for Fredholm integral equations of the first kind. Pitman, London, 1984
P.C. Hansen and D.P. O’Leary, The use of L-curve in the regularization of discrete ill-posed problems, SIAM Journal of Scientific Computing, Vol. 14, pp. 1487–1503, 1993
T. Regińska, A regularization parameter in discrete ill-posed problems, SIAM Journal Scientific Computing, Vol. 17, No. 3, pp. 740–749, 1996
D. Krawczyk-Stańdo and M. Rudnicki, Regularization parameter in discrete ill-posed problems-the use of the U-curve, International Journal of Applied Mathematics and Computer Science, Vol. 17, No. 2, pp. 101–108, 2007
P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, Vol. 34, No. 4, pp. 561–580, 1992
P. Neittaanmaki, M. Rudnicki, and A. Savini, Inverse Problems and Optimal Design in Electricity and Magnetism, Oxford, Clarendon Press, 1996
P.C. Hansen, Regularization tools, a Mat Lab Package for Analysis and Solution of Discrete Ill-Posed Problems – Report UNIC-92-03, 1993
D. Krawczyk-Stando and M. Rudnicki, Regularised synthesis of the magnetic field using the L-curve approach, International Journal of Applied Electromagnetics and Mechanics, Vol. 22, No. 3,4, pp. 233–242, 2005
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Krawczyk-Stańdo, D., Rudnicki, M. (2008). The Use of L-Curve and U-Curve in Inverse Electromagnetic Modelling. In: Wiak, S., Krawczyk, A., Dolezel, I. (eds) Intelligent Computer Techniques in Applied Electromagnetics. Studies in Computational Intelligence, vol 119. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78490-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-78490-6_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78489-0
Online ISBN: 978-3-540-78490-6
eBook Packages: EngineeringEngineering (R0)