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Newton regularizations for impedance tomography: a numerical study. (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:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
44A12Radon transform
78A25General electromagnetic theory
35Q60PDEs in connection with optics and electromagnetic theory
35R25Improperly posed problems for PDE
35R30Inverse problems for PDE
78M10Finite element methods (optics)