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 $x^†$ 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 $$ x_{k+1}^δ=x_0-g_{α_k} (F&#39;(x_k^δ)^*F&#39;(x_k^δ)) F&#39;(x_k^δ)^* (F(x_k^δ)-y^δ-F&#39;(x_k^δ)(x_k^δ-x_0)) $$ in the case that only available data is a noise $y^δ$ of $y$ satisfying $\|y^δ-y\|\le δ$ with a given small noise level $δ>0$. We terminate the iteration by the discrepancy principle in which the stopping index $k_δ$ is determined as the first integer such that $$ \|F(x_{k_δ}^δ)-y^δ\|\le τδ<\|F(x_k^δ)-y^δ\|, \qquad 0\le k<k_δ$$ with a given number $τ>1$. Under certain conditions on $\{α_k\}$, $\{g_α\}$ and $F$, we prove that $x_{k_δ}^δ$ converges to $x^†$ as $δ\to 0$ 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 Fréchet derivative $F&#39;$ of $F$ if $x_0-x^†$ is smooth enough.