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.