Newton Lavrentiev regularization for ill-posed operator equations in Hilbert scalese

Abstract

In this paper we consider the two step method for approximately solving the ill-posed operator equation F(x)=f, where F:D(F) subset of or equal to X to X, is a nonlinear monotone operator defined on a real Hilbert space X, in the setting of Hilbert scales. We derive the error estimates by selecting the regularization parameter alpha according to the adaptive method considered by Pereverzev and Schock in (2005), when the available data is f sup(delta) with ‖f-f sup(delta)‖ less than or equal to delta. The error estimate obtained in the setting of Hilbert scales {X sub(r)} sub(r are vectors in R generated by a densely defined, linear, unbounded, strictly positive self adjoint operator L:D(L) subset of X to X is of optimal order.