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 $x^{\mathrm{†}}$ 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 $$x_{k+1}^{\delta }=x_0-g_{{\alpha }_k}(F^{\mathrm{\prime }}(x_k^{\delta }{)}^{\ast }{F}^{\mathrm{\prime }}(x_k^{\delta }))F^{\mathrm{\prime }}(x_k^{\delta }{)}^{\ast }(F(x_k^{\delta })-y^{\delta }-F^{\mathrm{\prime }}(x_k^{\delta })(x_k^{\delta }-x_0))$$ in the case that only available data is a noise $y^{\delta }$ of $y$ satisfying $\mathrm{\parallel }{y}^{\delta }-y\mathrm{\parallel }\le \delta$ 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 $$\mathrm{\parallel }F(x_{k_{\delta }}^{\delta })-y^{\delta }\mathrm{\parallel }\le \tau \delta <\mathrm{\parallel }F(x_k^{\delta })-y^{\delta }\mathrm{\parallel },0\le k<k_{\delta }$$ with a given number $\tau >1$. Under certain conditions on $\{{\alpha }_k\}$, $\{g_{\alpha }\}$ and $F$, we prove that $x_{k_{\delta }}^{\delta }$ converges to $x^{\mathrm{†}}$ 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 $F^{\mathrm{\prime }}$ of $F$ if $x_0-x^{\mathrm{†}}$ is smooth enough.