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Zbl 1208.47058
Mahale, P.; Nair, M.T.
Iterated Lavrentiev regularization for nonlinear ill-posed problems. (English)
[J] ANZIAM J. 51, No. 2, 191-217 (2009). ISSN 1446-1811

The paper is concerned with stable approximation of a solution to a nonlinear ill-posed operator equation $F(x)=y$, where $F: X \to X$ is a monotone (in the sense of scalar product) differentiable operator in a real Hilbert space $X$. It is assumed that the element $y \in\Bbb R(F)$ is given by its $\delta$-approximation $y^{\delta}$, $\Vert y^{\delta}-y \Vert \leq \delta$. The iterative process $x_{k+1}=x_k-(F^{\prime}(x_k)+\alpha_k I)^{-1} [F(x_k)-y^{\delta}+\alpha_k (x_k-x_0)]$, $\alpha_k>0$, $\lim_{k \to \infty} \alpha_k=0$, is investigated. The authors propose an posteriori stopping rule $k=k(\delta,y^{\delta})$ and establish error estimates under the general source condition $x_0-x^*=\varphi(F'(x^*))v$, $F(x^*)=y$. The proofs use the additional assumption that $F^{\prime}(x^*)$ is selfadjoint.
[Mikhail Yu. Kokurin (Yoshkar-Ola)]

MSC 2000:: *47J06 Nonlinear ill-posed problems
65J20 Improperly posed problems (numerical methods in abstract spaces)
47J25 Methods for solving nonlinear operator equations (general)
65J15 Equations with nonlinear operators (numerical methods)

Keywords: Lavrentiev regularisation; nonlinear ill-posed problems; monotone operator; Fréchet derivative; regularisation parameter; stopping index; Gauss-Newton method; source function; order optimal estimate

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Zentralblatt MATH Berlin [Germany]