On a general regularization scheme for nonlinear ill-posed problems: II. Regularization in Hilbert scales

Abstract

In this paper we introduce a general regularization scheme to reconstruct solutions of nonlinear ill-posed problems where instead of y noisy data with are given and is a nonlinear operator between Hilbert spaces X and Y. In this regularization scheme regularized approximations are defined as a solution of the nonlinear equation where is a suitable initial guess, stands simply for and B denotes an unbounded self-adjoint strictly positive definite operator in the Hilbert space X. Assuming for and for some and ( is the norm in a Hilbert scale ) we prove that under certain conditions concerning the nonlinear operator F the regularized approximations satisfy the order optimal error bound provided that the function , the parameter s and the regularization parameter have been chosen properly. This paper extends earlier results where the special case s = 0 was treated.