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

Author: Jin Qi-nian
Journal: Math. Comp. 69 (2000), 1603-1623
MSC (1991): Primary 65J20, 45G10
Published electronically: February 18, 2000
MathSciNet review: 1680887
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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

\begin{displaymath}x_{k+1}^\delta=x_k^\delta-(\alpha_k I+F'(x_k^\delta)^*F'(x_k^... ...lta)^*(F(x_k^\delta)-y^\delta) +\alpha_k(x_k^\delta-x_0)\Big), \end{displaymath}

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

\begin{displaymath}\Vert x_{k_\delta}^\delta-x^\dag\Vert\le C\inf\Big\{\Vert x_k-x^\dag\Vert +\frac{\delta}{\sqrt{\alpha_k}}:k=0,1,\ldots\Big\} \end{displaymath}

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.

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Additional Information

Jin Qi-nian
Affiliation: Institute of Mathematics, Nanjing University, Nanjing 210008, P. R. China

Keywords: Nonlinear ill-posed problems, the iteratively regularized Gauss-Newton method, stopping rule, convergence, rates of convergence.
Received by editor(s): March 17, 1998
Received by editor(s) in revised form: January 4, 1999
Published electronically: February 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society