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

Abstract

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 {α _k }, {g _α } 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.