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

Recently, Vasin and George (2013) considered an iterative scheme for approximately solving an ill-posed operator equation . 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 ( is the initial guess and 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. 1. Introduction In this study, we are interested in approximately solving a nonlinear ill-posed operator equation: where is a nonlinear operator. Here is the domain of , and？ is the inner product with corresponding norm on the Hilbert spaces and . Throughout this paper we denote by the ball of radius centered at denotes the Fréchet derivative of at , and denotes the adjoint of . We assume that are the available noisy data satisfying where is the noise level. Equation (1) is, in general, ill-posed, in the sense that a unique solution that depends continuously on the data does not exist. Since the available data is , one has to solve (approximately) the perturbed equation instead of (1). To solve the ill-posed operator equations, various regularization methods are used, for example, Tikhonov regularization, Landweber iterative regularization, Levenberg-Marquardt method, Lavrentiev regularization, Newton type iterative method, and so forth (see, e.g., [1–16]). In [16], Vasin and George considered the iteration (which is a modified form of the method considered in [8]) where , is the initial guess, is the regularization parameter, and . Iteration (4) was used to obtain an approximation for the zero of and proved that is an approximate solution of (1). The regularization parameter in [16] was chosen appropriately from the finite set depending on the inexact data and the error level satisfying (2) using the adaptive parameter selection procedure suggested by Pereverzev and Schock [17]. In order to improve the rate of convergence many authors have considered the Hilbert scale variant of the regularization methods for solving ill-posed operator equations, for example, [18–26]. In this study, we present the Hilbert scale variant of (4). We consider the Hilbert scale (see [14, 18, 23, 26–29]) generated by a strictly positive self-adjoint operator , with the domain dense in satisfying , for all . Recall [19, 28] that the space is the