Stability in conductivity imaging from partial measurements of one interior current

Carlos Montalto; Alexandru Tamasan

doi:10.3934/ipi.2017016

Article Contents

2017, Volume 11, Issue 2: 339-353. Doi: 10.3934/ipi.2017016

This issue Previous Article Optical flow on evolving sphere-like surfaces Next Article Localization of the interior transmission eigenvalues for a ball

Stability in conductivity imaging from partial measurements of one interior current

Carlos Montalto^1, , and
Alexandru Tamasan^2,

1.
Deparment of Mathematics, University of Washingotn, Seattle, WA 98195-4350, USA
2.
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

* Corresponding author: Carlos Montalto
* Corresponding author: Carlos Montalto

Received: March 31, 2016

Revised: June 30, 2016

Published: May 2017

A. Tamasan was supported by the NSF Grant DMS 1312883.

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Abstract

We prove a stability result in the hybrid inverse problem of recovering the electrical conductivity from partial knowledge of one current density field generated inside a body by an imposed boundary voltage. The region of stable reconstruction is well defined by a combination of the exact and perturbed data. This work explains the high resolution and accuracy reconstructions in some existing numerical experiments that use partial interior data.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 35J65; Secondary: 65N21.

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Figure 2. The injectivity region $\mathcal{I}(\Gamma,u)$ is the light grey region that contains the stability region $\mathcal{S}(\Gamma,u)$ in dark grey.

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Figure 1. Illustration of the segment $l_p$ and the surface $\Sigma_p$.

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Figure 3. When $\Gamma$ is connected the visible region and the trajectory region can be the same.

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Figure 4. We illustrate how we can controlled the visible region and the projection of the current density. The underlying idea is to chose voltages potential $f$ at the boundary to induced specific level curves $u_0 =$const. By doing so, we get a better understanding of the required direction of the current density to obtain an stable reconstruction. Here, $\delta \bf{J}_0$ denote $\bf{J}(\sigma) - \bf{J}(\sigma_0)$ and ${{\bf{w}}_{0}} = \Pi_{\nabla (u_0)}(\delta \bf{J}_0) $.

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