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(x̂) = 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-x̂ (here x 0 is the initial approximation). The test example provided endorses the reliability and effectiveness of our method. © 2012 Springer-Verlag.