On a general regularization scheme for nonlinear ill-posed problems

Abstract

In this paper we introduce a general regularization scheme to reconstruct solutions of nonlinear ill-posed problems F(x)=y 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 solutions are defined as a fixed point of the nonlinear equation where and stands for . Assuming certain conditions concerning the nonlinear operator F and the smoothness of the element we derive stability estimates which show that the accuracy of the regularized solutions is order optimal provided that the function and the regularization parameter have been chosen properly.