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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−x̄) = yδ with some initial guess x̄. Assuming certain conditions concerning the nonlinear operator F and the smoothness of the element x̄−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.