Iteratively regularized methods for irregular nonlinear operator equations with a normally solvable derivative at the solution
The author considers a nonlinear equation \( F(x) = f \), where \( F: H_1 \to H_2 \) is a Fréchet differentiable operator between Hilbert spaces \( H_1 \) and \( H_2 \). In addition, the operator \( F^{\prime}(x^\ast) \) is assumed to be normally solvable at the desired solution \( x^\ast \in H_1 \) of the equation \( F(x) = f \), i.e., the range \( \mathcal{R}(F^{\prime}(x^\ast)) \) is a closed linear subspace of \( H_2 \). Let \( \tilde{f} \in H_2 \) be an approximately given right-hand side, with \( \| \tilde{f} - f \| \leq \delta \), where \( \delta > 0 \) is a known error level. The following class of iteratively regularized Gauß--Newton type methods is studied, \[ \begin{aligned} x_{n+1} = \xi - \Theta(F^{\prime}(x_n)^\ast F^{\prime}(x_n), \alpha_n) F^{\prime}(x_n)^\ast [ F(x_n) - \tilde{f} - F^{\prime}(x_n) (x_n - \xi)], \end{aligned} \] for \( n = 1,2,\dots \), where \( \xi \in H_1 \) is a fixed approximation to the solution \( x^\ast \), and \( x_0 \in H_1 \) is an initial approximation. In addition, \( \{ \alpha_n \} \) is a sequence of positive regularization parameters with \( \alpha_n \to 0 \) as \( n \to \infty \), and \( \Theta = \Theta(\lambda, \alpha) \) is a real-valued continuous function. A prominent example is given by \( \Theta(\lambda, \alpha) = (\lambda+ \alpha)^{-1} \), yielding the classical iteratively regularized Gauß-Newton method. In the noise-free case \( \delta = 0 \), i.e., \( \tilde{f} = f \), it is shown that \( \| x_n - x^\ast \| = \mathcal{O}(\alpha_n^p) \) holds as \( n \to \infty \), if \( \max_{\lambda \in I} | 1 - \Theta(\lambda,\alpha) \lambda | = \mathcal{O}(\alpha^{p}) \) as \( \alpha \to 0 \), and if several other conditions are satisfied, where \( p \geq 1 \), and \( I \) is some closed, bounded interval not containing the origin. In the case \( \delta > 0 \), an estimate \( \| x_{N(\delta)} - x^\ast \| = \mathcal{O}(\delta) \) as \( \delta \to 0 \) is obtained, where \( N = N(\delta) \) is determined either by a certain a priori parameter choice strategy or a discrepancy principle.
scientific article; zbMATH DE number 6701220
scientific article