A general convergence analysis on inexact Newton method for nonlinear inverse problems

We consider the inexact Newton methods $$x_{n+1}^\d=x_n^\d-g_{\a_n}(F'(x_n^\d)^* F'(x_n^\d)) F'(x_n^\d)^* (F(x_n^\d)-y^\d)$$ for solving nonlinear ill-posed inverse problems $F(x)=y$ using the only available noise data $y^\d$ satisfying $\|y^\d-y\|\le \d$ with a given small noise level $\d>0$. We terminate the iteration by the discrepancy principle $$\|F(x_{n_\d}^\d)-y^\d\|\le \tau \d<\|F(x_n^\d)-y^\d\|, \qquad 0\le n<n_\d$$ with a given number $\tau>1$. Under certain conditions on $\{\a_n\}$ and $F$, we prove for a large class of spectral filter functions $\{g_\a\}$ the convergence of $x_{n_\d}^\d$ to a true solution as $\d\rightarrow 0$. Moreover, we derive the order optimal rates of convergence when certain H\"{o}lder source conditions hold. Numerical examples are given to test the theoretical results.