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Modern regularization techniques for inverse modelling: a comparative study

Krawczyk-Stańdo D., Rudnicki M., Stańdo J.

Journal of Applied Computer Science

2011

Vol. 19, nr 1

51-63

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.

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

Krawczyk-Stańdo D., Rudnicki M.

International Journal of Applied Mathematics and Computer Science

2007

Vol. 17, no 2

157-164

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.