An application of newton type iterative method for lavrentiev regularization for ill-posed equations: Finite dimensional realization

Abstract

In this paper, we consider, a finite dimensional realization of Newton type iterative method for Lavrentiev regularization of ill-posed equations. Precisely we consider the ill-posed equation F(x) = f when the available data is f ? with

f - f ?

? ? and the operator F: D(F) ? X ? X is a nonlinear monotone operator defined on a real Hilbert space X. The error estimate obtained under a general source condition on x 0 - x? (where x 0 is the initial guess and x? is the solution of F(x) = f) is of optimal order. The regularization parameter ? is chosen according to the adaptive method considered by Perverzev and Schock (2005). An example is provided to show the efficiency of the proposed method.