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Tikhonov replacement functionals for iteratively solving nonlinear operator equations. (English) Zbl 1078.47030
The paper is concerned with an ill-posed nonlinear equation $$F(x)=y$$, where $$F: X \to Y$$ is a twice continuously differentiable operator between Hilbert spaces $$X$$ and $$Y$$. Let $$y^{\delta} \in Y$$ be an available approximation to $$y$$ with $$\| y^{\delta}-y \| \leq \delta$$. The authors suggest a two-level Tikhonov-type regularization method $$x_{k+1}=\text{argmin}_{x \in X}{\widetilde J}_{\alpha}(x,x_k)$$, $$k=0,1, \dots$$, $$\alpha \to 0$$, where $${\widetilde J}_{\alpha}(x,a)=J_{\alpha}(x)+ C\| x-a\|^2- \| F(x)-F(a)\|^2$$, $$C, \alpha>0$$, and $$J_{\alpha}(x)=\| y^{\delta}-F(x)\|^2+\alpha \| x-{\overline x}\|^2$$ is the classical Tikhonov functional. Let the derivative $$F^{\prime}(x)$$ be Lipschitz continuous on $$X$$ with a constant $$L$$. It is shown that the modified functional $${\widetilde J}_{\alpha}(x,x_k)$$ is strictly convex and that $$x_{k+1}$$ can be obtained by the fixed point iteration, provided that $$L \sqrt{J_{\alpha}(a)}<C+\alpha$$. Moreover, if $$\| F(x)-F({\widetilde x})-F^{\prime}({\widetilde x})(x-{\widetilde x})\| \leq \| F(x)-F({\widetilde x})\|$$ $$\forall x, {\widetilde x} \in X$$ and the solution $$x^*$$ to the original equation satisfies $$x^*-{\overline x} \in R(F^{\prime *}(x^*))$$, then the method with an appropriate stopping rule $$\alpha=\alpha(\delta)$$ generates approximations converging to $$x^*$$ as $$\delta \to 0$$.

##### MSC:
 47J06 Nonlinear ill-posed problems 47J25 Iterative procedures involving nonlinear operators
TIGRA
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