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 $${x^\dagger}$$ 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$$x_{k+1}^{\delta}=x_{0}-g_{\alpha_{k}}\left(F'(x_{k}^{\delta})^*F'(x_{k}^{\delta})\right) F'(x_{k}^{\delta})^*\left(F(x_{k}^{\delta})-y^{\delta}-F'(x_{k}^{\delta})(x_{k}^{\delta}-x_{0})\right)$$in the case that only available data is a noise $${y^\delta}$$ of y satisfying $${\|y^\delta - y\| \le \delta}$$ with a given small noise level $${\delta > 0}$$ . We terminate the iteration by the discrepancy principle in which the stopping index $${k_\delta}$$ is determined as the first integer such that$$\|F(x_{k_\delta}^{\delta})-y^{\delta}\|\le \tau \delta < \|F(x_{k}^{\delta})-y^{\delta}\|, \quad 0\le k < k_{\delta}$$with a given number τ > 1. Under certain conditions on {αk}, {gα} and F, we prove that $${x_{k_\delta}^{\delta}}$$ converges to $${x^\dagger}$$ as $${\delta \rightarrow 0}$$ 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 $${x_{0} - x^\dagger}$$ is smooth enough.