On convergence of regularized modified Newton's method for nonlinear ill-posed problems

Abstract

Abstract In this paper we consider regularized modified Newton's method for approximately solving the nonlinear ill-posed problem F(x) = y, where the right hand side is replaced by noisy data yδ ∈ Y with ‖y – yδ ‖ ≤ δ and F : D(F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y. Under the assumption that Fréchet derivative F′ of F is Lipschitz continuous, a choice of the regularization parameter and a stopping rule based on a majorizing sequence are presented. We prove that under a general source condition on , the error between the regularized approximation and the solution of optimal order.