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

Kha Van Huynh; Barbara Kaltenbacher

doi:10.3934/ipi.2020074

Article Contents

2021, Volume 15, Issue 3: 415-443. Doi: 10.3934/ipi.2020074

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

Kha Van Huynh and
Barbara Kaltenbacher^,

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

^* Corresponding author: Barbara Kaltenbacher
^* Corresponding author: Barbara Kaltenbacher

Received: March 31, 2020

Revised: July 31, 2020

Early access: November 2020

Published: June 2021

supported by the Austrian Science Fund FWF under grant P30054

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Abstract

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.

Keywords:

inverse problems for PDEs,
regularization,
electrical impedance tomography,
identification of magnetic permeability,
localization of sound sources.

Mathematics Subject Classification: Primary: 65J20; 35R30; Secondary: 35Q60.

Citation:

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Figure 1. Electrodes (in red) on the boundary with $ L = 4 $

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Figure 2. Typical $ B $–$ H $-curve shape (left) and measurement setup (right)

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