Projection Scheme for Newton-Type Iterative Method for Lavrentiev Regularization

Abstract

In this paper we consider the finite dimensional realization of a Newton-type iterative method for obtaining an approximate solution to the nonlinear ill-posed operator equation F(x) = f, where F:D(F) ⊆ X → X is a nonlinear monotone operator defined on a real Hilbert space X. It is assumed that \(F(\hat{x})=f\) and that the only available data are f δ with ∥ f − f δ ∥ ≤ δ. It is proved that the proposed method has a local convergence of order three. The regularization parameter α is chosen according to the balancing principle considered by Perverzev and Schock (2005) and obtained an optimal order error bounds under a general source condition on \(x_0-\hat{x}\) (here x 0 is the initial approximation). The test example provided endorses the reliability and effectiveness of our method.