An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems

In this paper we consider the Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations F(x) = y where F : D(F) subset of X -> X is a nonlinear monotone operator or F'(x(0)) is nonnegative selfadjoint operator defined on a real Hilbert space X. We assume that only a noisy data y delta is an element of X with parallel to y - y(delta)parallel to <= delta d are available. Further we assume that Frechet derivative F' of F satisfies center-type Lipschitz condition. A priori choice of regularization parameter is presented. We proved that under a general source condition on x(0) - (x) over cap; the error parallel to(x) over cap - x(n,alpha)(delta)parallel to between the regularized approximation x(n,alpha)(delta)( x(n,alpha)(delta) := x(0)) and the solution (x) over cap is of optimal order. In the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem. (C) 2013 Elsevier Inc. All rights reserved.