Modern regularization techniques for inverse modelling: a comparative study

• Autorzy: Krawczyk-Stańdo D. , Rudnicki M. , Stańdo J.
• Języki publikacji: EN

Abstrakty

EN

Regularization techniques are used for computing stable solutions to ill-posed problems. The well-known form of regularization is that of Tikhonov in which the regularized solution is searched as a minimiser of the weighted combination of the residual norm and a side constraint-controlled by the regularization parameter. For the practical choice of regularization parameter we can use the L-curve approach, U-curve criterion introduced by us [1] and empirical risk method [2]. We present a comparative study of different strategies for the regularization parameter choice on examples of function approximation by radial basis neural networks. Such networks are universal approximators and can learn any nonlinear mapping. e.g. representing an magnetic inverse problem. Some integral equations of the first kind are considered as well.

Słowa kluczowe

EN

regularization parameter; L-curve; U-curve; discrepancy principle; empirical risk; inverse problem

PL

parametr regularyzacji; krzywa L; krzywa U; zasada rozbieżności; ryzyko empiryczne; zagadnienie odwrotne

• Wydawca: Wydawnictwo Politechniki Łódzkiej
• Czasopismo: Journal of Applied Computer Science
• Rocznik: 2011
• Tom: Vol. 19, nr 1
• Strony: 51--63
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Krawczyk-Stańdo D.

autor

Rudnicki M.

autor

Stańdo J.

• Technical University of Łódź, Centre of Mathematics and Physics, 90-924 Łódź, al. Politechniki 11, Poland,

Bibliografia

• [1] Krawczyk-Stańdo, D. and Rudnicki, M., 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, 2007, pp. 101-108.
• [2] Vapnik, S., Estimation of dependencies based on empirical data, Verlag, Berlin, 1982.
• [3] Krawczyk-Stańdo, D. and Rudnicki, M., Regularised synthesis of the magnetic field using the L-curve approach, International Journal of Applied Electromagnetics and Mechanics, Vol. 22, No. 3,4, 2005, pp. 233-242.
• [4] Hansen, P. C. and O'Leary, D. P., The use of the L-curve in the regularization of discrete ill- posed problems, SIAM J. Sci. Comput., Vol. 14, 1993, pp. 1487-1503.
• [5] D. Krawczyk-Stańdo, M. R. and Stańdo, J., Radial neural network learning using U-curve approach, Polish Journal of Environmental Studies, Vol. 17s, No. 2A, 2008, pp. 42-46.
• [6] Haykin, S., Neural Networks, a Comprehensive Foundation, Macmillan College Publishing Company, New York, 1994.
• [7] Hansen, P. C., Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems; version 2.0 for Matlab 4.0, Report UNIC-92-03, 1993.