An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems

Abstract

In this paper we consider the Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations F(x)=y where F:D(F)⊆X⟶X is a nonlinear monotone operator or F′(x0) is nonnegative selfadjoint operator defined on a real Hilbert space X. We assume that only a noisy data yδ∈X with ‖y-yδ‖⩽δ are available. Further we assume that Fréchet derivative F′ of F satisfies center-type Lipschitz condition. A priori choice of regularization parameter is presented. We proved that under a general source condition on x0-xˆ, the error ‖xˆ-xn,αδ‖ between the regularized approximation xn,αδ(x0,αδ≔x0) and the solution xˆ is of optimal order. In the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem.