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

摘要

In this paper we consider the two step method for approximately solving the ill-posed operator equation F(x)=f, where F:D(F)⊆X→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 α according to the adaptive method considered by Pereverzev and Schock in (2005), when the available data is fδ with ‖f-fδ‖⩽δ. The error estimate obtained in the setting of Hilbert scales {Xr}r∈R generated by a densely defined, linear, unbounded, strictly positive self adjoint operator L:D(L)⊂X→X is of optimal order.