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Zbl 1129.65043
Langer, S.; Hohage, T.
Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions.
(English)
[J] J. Inverse Ill-Posed Probl. 15, No. 3, 311-327 (2007). ISSN 0928-0219; ISSN 1569-3945/e

The authors consider the nonlinear ill-posed operator equation $F(x) =y$, where $F: D(F)\rightarrow \text{\it Y}$ is injective and continuously Fréchet differentiable on its domain $D(F) \subset X$; $X$, $Y$ are Hilbert spaces. They assume that there exists a solution $x^{\dag}$ of the equation, and that only noisy data $y^{\delta}$ satisfying $\Vert y^{\delta}-y \Vert \leq\delta$ are available. To iteratively compute an approximation to $x^{\dag}$ they replace the $n$-th Newton step by the linearized equation $$F'[x^{\delta}_{n}]h_{n}=y^{\delta}-F(x^{\delta}_{n}), n=0,1,2,\dots$$ where $h_{n}= x^{\delta}_{n+1}-x^{\delta}_{n}$. Since the linearized equation inherits the ill-posedness of the initial equation the authors apply a Tikhonov regularization with the initial guess $x_{0}-x^{\delta}_{n}$, called the regularized Gauss-Newton method (IRGNM). The following problems are studied: Convergence of the IRGNM for exact data; IRGNM with discrepancy principle for nonlinear problems; Solving the linearized equation.
[Erwin Schechter (Moers)]
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
65J20 Improperly posed problems (numerical methods in abstract spaces)
65J22 Inverse problems
47J06 Nonlinear ill-posed problems

Keywords: nonlinear inverse problems; regularized Newton methods; source conditions; convergenge rates; nonlinear ill-posed operator equation; Hilbert spaces; Tikhonov regularization; Gauss-Newton method; discrepancy principle

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