Further convergence results on the general iteratively regularized Gauss-Newton methods under the discrepancy principle

Abstract:

We consider the general iteratively regularized Gauss-Newton methods \[ x_{k+1}^\delta =x_0-g_{\alpha _k}(F’(x_k^\delta )^*F’(x_k^\delta )) F’(x_k^\delta )^* \left (F(x_k^\delta )-y^\delta -F’(x_k^\delta )(x_k^\delta -x_0)\right ) \] for solving nonlinear inverse problems $F(x)=y$ using the only available noise $y^\delta$ of $y$ satisfying $\|y^\delta -y\|\le \delta$ with a given small noise level $\delta >0$. In order to produce reasonable approximation to the sought solution, we terminate the iteration by the discrepancy principle. Under much weaker conditions we derive some further convergence results which improve the existing ones and thus expand the applied range.