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 F(x)=y when the data y y is given approximately by y δ y^\delta with ‖ y δ − y ‖ ≤ δ \|y^\delta -y\|\le \delta . In this method, the iterative sequence { x k δ } \{x_k^\delta \} is defined successively by \[ x k + 1 δ = x k δ − ( α k I + F ′ ( x k δ ) ∗ F ′ ( x k δ ) ) − 1 ( F ′ ( x k δ ) ∗ ( F ( x k δ ) − y δ ) + α k ( x k δ − x 0 ) ) , x_{k+1}^\delta =x_k^\delta -(\alpha _k I+F’(x_k^\delta )^*F’(x_k^\delta )) ^{-1}\Big (F’(x_k^\delta )^*(F(x_k^\delta )-y^\delta ) +\alpha _k(x_k^\delta -x_0)\Big ), \] where x 0 δ := x 0 x_0^\delta :=x_0 is an initial guess of the exact solution x † x^\dagger and { α k } \{\alpha _k\} is a given decreasing sequence of positive numbers admitting suitable properties. When x k δ x_k^\delta is used to approximate x † 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 δ k_\delta determined by this rule it is proved that \[ ‖ x k δ δ − x † ‖ ≤ C inf { ‖ x k − x † ‖ + δ α k : k = 0 , 1 , … } \|x_{k_\delta }^\delta -x^\dagger \|\le C\inf \Big \{\|x_k-x^\dagger \| +\frac {\delta }{\sqrt {\alpha _k}}:k=0,1,\ldots \Big \} \] with a constant C C independent of δ \delta , where x k x_k denotes the iterative solution corresponding to the noise free case. As a consequence of this result, the convergence of x k δ δ x_{k_\delta }^\delta is obtained, and moreover the rate of convergence is derived when x 0 − x † 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 > ν ≤ 1 0>u \le 1 . Numerical examples for parameter estimation of a differential equation are given to test the theoretical results.