On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems

The iteratively regularized Gauss-Newton method is applied to compute the stable solutions to nonlinear ill-posed problems $F(x)=y$ when the data $y$ is given approximately by $y^\delta$with $\Vert y^\delta-y\Vert\le\delta$. In this method, the iterative sequence $\{x_k^\delta\}$ is defined successively by

where $x_0^\delta:=x_0$ is an initial guess of the exact solution $x^\dag$ and $\{\alpha_k\}$ is a given decreasing sequence of positive numbers admitting suitable properties. When $x_k^\delta$ is used to approximate $x^\dag$, the stopping index should be designated properly. In this paper, an a posteriori stopping rule is suggested to choose the stopping index of iteration, and with the integer $k_\delta$ determined by this rule it is proved that

with a constant $C$ independent of $\delta$, where $x_k$ denotes the iterative solution corresponding to the noise free case. As a consequence of this result, the convergence of $x_{k_\delta}^\delta$is obtained, and moreover the rate of convergence is derived when $x_0-x^\dag$ satisfies a suitable ``source-wise representation". The results of this paper suggest that the iteratively regularized Gauss-Newton method, combined with our stopping rule, defines a regularization method of optimal order for each $0<\nu\le 1$. Numerical examples for parameter estimation of a differential equation are given to test the theoretical results.