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 $x^\dag$ of nonlinear ill-posed inverse problems $F(x)=y$ with nonlinear operators $F:X\to Y$ between two Hilbert spaces $X$ and $Y$ by the Newton type methods $$ x_{k+1}^\delta=x_0-g_{\alpha_k} (F'(x_k^\delta)^*F'(x_k^\delta)) F'(x_k^\delta)^* (F(x_k^\delta)-y^\delta-F'(x_k^\delta)(x_k^\delta-x_0)) $$ 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\|, \qquad 0\le k 1$. Under certain conditions on $\{\alpha_k\}$, $\{g_\alpha\}$ and $F$, we prove that $x_{k_\delta}^\delta$ converges to $x^\dag$ as $\delta\to 0$ 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 Fr\'{e}chet derivative $F'$ of $F$ if $x_0-x^\dag$ is smooth enough.