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Stability in conductivity imaging from partial measurements of one interior current

  • * Corresponding author: Carlos Montalto

    * Corresponding author: Carlos Montalto 
A. Tamasan was supported by the NSF Grant DMS 1312883.
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  • 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.

    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.

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

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

    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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