zbMATH — the first resource for mathematics

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 \(\delta \to 0\).

47J06 Nonlinear ill-posed problems
47J25 Iterative procedures involving nonlinear operators
PDF BibTeX Cite
Full Text: DOI