*(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 FrĂ©chet differentiable on its domain $D\left(F\right)\subset X$; $X$, $Y$ are Hilbert spaces. They assume that there exists a solution ${x}^{\u2020}$ 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 ${x}^{\u2020}$ they replace the $n$-th Newton step by the linearized equation

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 |