Newton Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems in Hilbert Scales

Abstract

Recently, Vasin and George (2013) considered an iterative scheme for approximately solving an ill-posed operator equation F(x)=y. In order to improve the error estimate available by Vasin and George (2013), in the present paper we extend the iterative method considered by Vasin and George (2013), in the setting of Hilbert scales. The error estimates obtained under a general source condition on x0-x^ (x0 is the initial guess and x^ is the actual solution), using the adaptive scheme proposed by Pereverzev and Schock (2005), are of optimal order. The algorithm is applied to numerical solution of an integral equation in Numerical Example section. 2014 Monnanda Erappa Shobha and Santhosh George.