On the method of Lavrentiev regularization for nonlinear ill-posed problems

In this paper we study the method of Lavrentiev regularization to reconstruct solutions x(dagger) of nonlinear ill-posed problems F(x) = y where instead of y noisy data y(delta) is an element of X with \\y - y(delta) \\ less than or equal to delta are given and F : D (F) subset of X --> X is a monotone nonlinear operator. In this regularization method regularized solutions x(alpha)(delta) are obtained by solving the singularly perturbed nonlinear operator equation F(x) + alpha(x - (x) over bar) with some initial guess (x) over bar. Assuming certain conditions concerning the nonlinear operator F and the smoothness of the element (x) over bar - x(dagger) we derive stability estimates which show that the accuracy of the regularized solutions is order optimal provided that the regularization parameter a has been chosen properly.