A general convergence analysis of some newton-type methods for nonlinear inverse problems

We consider the methods x_n+1^δ = x _n^δ-g_αn (F′(x_n ^δ)*F′( x_n^δ))F′( x_n^δ)′(F(x_n^δ)- y^δ) for solving nonlinear ill-posed inverse problems F(x) = y using the only available noise data y^δ satisfying ∥y ^δ - y∥ ≤ δ with a given small noise level δ >0. We terminate the iteration by the discrepancyprinciple ∥F(x _nδ^δ)-y^δ ∥ ≤ τ^δ ≤ ∥F(x_n^δ)-y ^δ ∥, 0 ≤ n < n_δ, with a given number τ > 1. Under certainconditions on {α_n} and F, we prove for a large class of spectral filter functions {gα} the convergenceof x_nδ^δ to a true solution as δ → 0. Moreover, we derive the order optimal rates of convergence when certain Hölder source conditions hold. Numerical examples are given to test the theoretical results.