Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields

Chenxi Guo; Guillaume Bal

doi:10.3934/ipi.2014.8.1033

Article Contents

2014, Volume 8, Issue 4: 1033-1051. Doi: 10.3934/ipi.2014.8.1033

This issue Previous Article A real-time D-bar algorithm for 2-D electrical impedance tomography data Next Article A data-driven edge-preserving D-bar method for electrical impedance tomography

Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields

Chenxi Guo^1, and
Guillaume Bal^2,

1.
Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027
2.
Department of Applied Physics and Applied Mathematics, Columbia University, 200 S. W. Mudd Building, MC 4701, 500 W. 120th Street, New York, NY 10027

Received: July 31, 2013

Revised: April 30, 2014

Published: October 2014

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Abstract

This paper concerns the reconstruction of a complex-valued anisotropic tensor $\gamma = \sigma + \iota\omega\varepsilon$ from knowledge of several internal magnetic fields $H$, where $H$ satisfies the anisotropic Maxwell system on a bounded domain with prescribed boundary conditions. We show that $\gamma$ can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition of $H$. A minimum number of $6$ such functionals is sufficient to obtain a local reconstruction of $\gamma$ in dimension three provided that the electric field satisfies appropriate boundary conditions. When $\gamma$ is close to a scalar tensor, such boundary conditions are shown to exist using the notion of complex geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a Runge approximation property is used instead to obtain partial results. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.

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Mathematics Subject Classification: Primary: 35R30, 35S05; Secondary: 35J47.

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