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On the use of fixed point iterations for the regularization of nonlinear ill-posed problems. (English) Zbl 1077.65058
The paper is concerned with a nonlinear ill-posed operator equation \(F(x)=y\) defined by a twice continuously differentiable operator \(F\) in a Hilbert space. Let \(y^{\delta}\) be an approximation to the exact measurement \(y\) and \(\| y^{\delta}-y \| \leq \delta\). For solving the problem the author suggests a two-stage iterative process: starting from \(x_{\alpha_{k-1}}^{\delta}\) compute \(x_{\alpha_{k}}^{\delta}\) as a fixed point of a contraction \(R_{\alpha_k}(x,x_{\alpha_{k-1}}^{\delta})\) and repeat with \(\alpha_{k+1}=q \alpha_k\), \(q \in (0,1)\) until \(\| F(x_{\alpha_{k}}^{\delta})-y^{\delta} \| \leq c\delta\). The mapping \(R_{\alpha}(x,x_{\bar{\alpha}}^{\delta})\) involves second derivatives of the operator \(F\). Under the assumption that the solution \(x^*\) possesses a sourcewise representation, the algorithm terminates within a finite number \(k^*\) of outer iterations and \(\| x_{\alpha_{k^*}}^{\delta} -x^*\|={\mathcal O}(\sqrt {\delta})\).

MSC:
65J15 Numerical solutions to equations with nonlinear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47J06 Nonlinear ill-posed problems
Software:
TIGRA
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