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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
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