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† of nonlinear ill-posed problems F (x) = y where instead of y noisy data yδ ∊ X with || y − yδ|| ≤ δ are given and F : D(F) ⊂ X → X is a monotone nonlinear operator. In this regularization method regularized solutions xαδ are obtained by solving the singularly perturbed nonlinear operator equation F (x) + α(x−) = yδ with some initial guess . Assuming certain conditions concerning the nonlinear operator F and the smoothness of the element −x† we derive stability estimates which show that the accuracy of the regularized solutions is order optimal provided that the regularization parameter α has been chosen properly.