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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 $F\left(x\right)=y$, where $F:D\left(F\right)\to \mathit{\text{Y}}$ is injective and continuously Fréchet differentiable on its domain $D\left(F\right)\subset X$; $X$, $Y$ are Hilbert spaces. They assume that there exists a solution ${x}^{†}$ of the equation, and that only noisy data ${y}^{\delta }$ satisfying $\parallel {y}^{\delta }-y\parallel \le \delta$ are available. To iteratively compute an approximation to ${x}^{†}$ they replace the $n$-th Newton step by the linearized equation

${F}^{\text{'}}\left[{x}_{n}^{\delta }\right]{h}_{n}={y}^{\delta }-F\left({x}_{n}^{\delta }\right),n=0,1,2,\cdots$

where ${h}_{n}={x}_{n+1}^{\delta }-{x}_{n}^{\delta }$. Since the linearized equation inherits the ill-posedness of the initial equation the authors apply a Tikhonov regularization with the initial guess ${x}_{0}-{x}_{n}^{\delta }$, 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.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 65J20 Improperly posed problems; regularization (numerical methods in abstract spaces) 65J22 Inverse problems (numerical methods in abstract spaces) 47J06 Nonlinear ill-posed problems