*(English)*Zbl 1154.65042

A nonlinear ill-posed operator equation $F\left(x\right)=y$, from the domain $D\left(F\right)\subseteq X$ of the Hilbert space $X$ into a Hilbert space $Y$, is considered in case of only noisy data ${y}^{\delta}$ are available, with the assumption $\parallel y-y\parallel \le \delta $, $\delta >0$. The Tikhonov regularization method consists in using the Tikhonov functional

The convergence rates for this type of regularization under a mild regularity assumption on the solution, namely source conditions of logarithmic type, are proved. For the choice of the regularization parameter a priori or a posteriori strategies according to the discrepancy can be used. Restrictions on the nonlinearity of the forward operator are made unless the initial error is sufficiently smooth.

##### MSC:

65J15 | Equations with nonlinear operators (numerical methods) |

65J20 | Improperly posed problems; regularization (numerical methods in abstract spaces) |

47J06 | Nonlinear ill-posed problems |