# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2020074

## Some application examples of minimization based formulations of inverse problems and their regularization

 Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria

* Corresponding author: Barbara Kaltenbacher

Received  April 2020 Revised  August 2020 Published  November 2020

Fund Project: supported by the Austrian Science Fund FWF under grant P30054

In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems.

Citation: Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, doi: 10.3934/ipi.2020074
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Electrodes (in red) on the boundary with $L = 4$
Typical $B$$H$-curve shape (left) and measurement setup (right)