Fractional Regularization Methods For Ill-Posed Problems In Hilbert Scales

Abstract

Regularization methods are widely used for solving ill-posed problems. In this thesis we consider the fractional regularization methods for solving equations of the form T (x) = y where T : X −→ Y is a linear operator between the Hilbert spaces X and Y . In practical applications, we mostly have only the noisy data yδ such that ‖y − yδ ‖ ≤ δ. Throughout our study, we work in the setting of Hilbert scales as it improves the order of convergence. We study the Fractional Tikhonov Regularization method in Hilbert scales and for selecting the regulariza- tion parameter the adaptive choice method introduced by Pereverzev and Schock (2005) is used. Also we introduce a new parameter choice strategy. While per- forming numerical calculations, it is always easier to work in a finite dimensional setting than in an infinite dimensional one. For this reason we study the finite dimensional realization of the fractional Tikhonov and the fractional Lavrentiev regularization methods in Hilbert scales. We also study the analogous of the dis- crepancy principle considered in George and Nair (1993) for Fractional Lavrentiev method.