Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

We present a new iterative method which does not involve inversion of the operators for obtaining an approximate solution for the nonlinear ill-posed operator equation . The proposed method is a modified form of Tikhonov gradient (TIGRA) method considered by Ramlau (2003). The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005). The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper. 1. Introduction This paper is devoted to the study of nonlinear ill-posed problem where is a nonlinear operator between the Hilbert spaces and . We assume that is a Fréchet-differentiable nonlinear operator acting between infinite dimensional Hilbert spaces and with corresponding inner products and norms , respectively. Further it is assumed that (1) has a solution , for exact data; that is, , but due to the nonlinearity of this solution need not be unique. Therefore we consider a -minimal norm solution of (1). Recall that [1–3] a solution of (1) is said to be an -minimal norm ( -MNS) solution of (1) if In the following, we always assume the existence of an -MNS for exact data . The element in (3) plays the role of a selection criterion [4] and is assumed to be known. Since (1) is ill-posed, regularization techniques are required to obtain an approximation for . Tikhonov regularization has been investigated by many authors (see e.g., [2, 4, 5]) to solve nonlinear ill-posed problems in a stable manner. In Tikhonov regularization, a solution of the problem (1) is approximated by a solution of the minimization problem where is a small regularization parameter and is the available noisy data, for which we have the additional information that It is known [3] that the minimizer of the functional satisfies the Euler equation of the Tikhonov functional . Here denotes the adjoint of the Fréchet derivative . It is also known that [1] for properly chosen regularization parameter , the minimizer of the functional is a good approximation to a solution with minimal distance from . Thus the main focus is to find a minimizing element of the Tikhonov functional (4). But the Tikhonov functional with nonlinear operator might have several minima, so to ensure the convergence of any optimization algorithm to a global minimizer of the Tikhonov functional (3), one has to employ stronger restrictions on the operator [6]. In the last few years many authors considered iterative methods, for example, Landweber method [7, 8],