On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

We consider the computation of stable approximations to the exact solution x^† of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods x_k+1^δ=x₀-g _αk(F'(x_k^δ) ^*F'(x_k^δ) F'(x_k ^δ)^*(F(x_k^δ)-y ^δ-F'(x_k^δ)(x_k ^δ-x₀) in the case that only available data is a noise y^δ of y satisfying ||y^δ- y|| ≤ δ with a given small noise level δ > 0. We terminate the iteration by the discrepancy principle in which the stopping index k _δ is determined as the first integer such that ||F(x _kδ^δ)-y^δ||≤ τ δ < ||F(x_k^δ)-y^δ||, 0≤ k < k_δ with a given number τ > 1. Under certain conditions on {α _k }, {g _α } and F, we prove that x_kδ^δ converges to x ^† as δ → 0 and establish various order optimal convergence rate results. It is remarkable that we even can show the order optimality under merely the Lipschitz condition on the Fréchet derivative F' of F if x₀ - x^† is smooth enough.