International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 39, Pages 2487-2499
doi:10.1155/S0161171203203197
An optimal order yielding discrepancy principle for simplified regularization of ill-posed problems in Hilbert scales
1Department of Mathematics, Government College, Sanquelim 403505, Goa, India
2Department of Mathematics, Indian Institute of Technology, Madras 600 036, Chennai, India
Received 19 March 2002
Copyright © 2003 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
Recently, Tautenhahn and Hämarik (1999) have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equation Tx=y, where T is a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of Hilbert scales, when T is a positive and selfadjoint operator. When the data y is known only approximately, our method provides optimal order under certain natural assumptions on the ill-posedness of the equation and smoothness of the solution. The result, in fact, improves an earlier work of the authors (1997).