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

Abstract:

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 $\|y^\delta -y\|\le \delta$. In this method, the iterative sequence $\{x_k^\delta \}$ is defined successively by \[ x_{k+1}^\delta =x_k^\delta -(\alpha _k I+F’(x_k^\delta )^*F’(x_k^\delta )) ^{-1}\Big (F’(x_k^\delta )^*(F(x_k^\delta )-y^\delta ) +\alpha _k(x_k^\delta -x_0)\Big ), \] where $x_0^\delta :=x_0$ is an initial guess of the exact solution $x^\dagger$ and $\{\alpha _k\}$ is a given decreasing sequence of positive numbers admitting suitable properties. When $x_k^\delta$ is used to approximate $x^\dagger$, 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 \[ \|x_{k_\delta }^\delta -x^\dagger \|\le C\inf \Big \{\|x_k-x^\dagger \| +\frac {\delta }{\sqrt {\alpha _k}}:k=0,1,\ldots \Big \} \] 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 $x_0-x^\dagger$ 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.