International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 37, Pages 1973-1996
doi:10.1155/S0161171204306095
Optimal order yielding discrepancy principle for simplified regularization in Hilbert scales: finite-dimensional realizations
1Department of Mathematics, Government College of Arts, Science and Commerce, Sanquelim, Goa 403505, India
2Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
Received 13 June 2003
Copyright © 2004 Santhosh George and M. Thamban Nair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Simplified regularization using finite-dimensional approximations in the setting of Hilbert scales has been considered for obtaining stable approximate solutions to ill-posed operator equations. The derived error estimates using an a priori and a posteriori choice of parameters in relation to the noise level are shown to be of optimal order with respect to certain natural assumptions on the ill posedness of the equation. The results are shown to be applicable to a wide class of spline approximations in the setting of Sobolev scales.