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Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions. (English) Zbl 1129.65043
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.

65J15Equations with nonlinear operators (numerical methods)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
65J22Inverse problems (numerical methods in abstract spaces)
47J06Nonlinear ill-posed problems
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