*(English)*Zbl 1109.65100

Summary: The inexact Newton iteration REGINN for regularizing nonlinear ill-posed problems consists of two components: the (outer) Newton iteration, stopped by a discrepancy principle, and the inner iteration, which computes the Newton update by solving approximately a linearized system. The second author proved the convergence of REGINN furnished with the conjugate gradient method as inner iteration [cf. *A. Rieder*, SIAM J. Numer. Anal. 43, No. 2, 604–622 (2005; Zbl 1092.65047)].

Amongst others the following feature distinguishes REGINN from other Newton-like regularization schemes: the regularization level for the locally linearized systems can be adapted dynamically incorporating information on the local degree of ill-posedness gained during the iteration. Of course, the potential of this feature can be fully explored only by meaningful numerical experiments in a realistic setting.

Therefore, we apply REGINN to the 2D-electrical impedance tomography problem with the complete electrode model. This inverse problem is known to be severely ill-posed. The achieved reconstructions are compared qualitatively and quantitatively with reconstructions from a one-step method which is closely related to the NOSER algorithm [cf. *M. Cheney, D. Isaacson, D. Newell, S. Simske* and *J. Goble*, Int. J. Imag. Syst. Technol. 2, 66–75 (1990)], an often used solver in impedance tomography. Our detailed numerical comparison reveals REGINN to be a competitive solver outperforming the one-step method when noise corrupts the data and/or a moderately large number of electrodes is used.

##### MSC:

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

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

44A12 | Radon transform |

78A25 | General electromagnetic theory |

35Q60 | PDEs in connection with optics and electromagnetic theory |

35R25 | Improperly posed problems for PDE |

35R30 | Inverse problems for PDE |

78M10 | Finite element methods (optics) |