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Towards a \(d\)-bar reconstruction method for three-dimensional EIT. (English) Zbl 1111.35108

Summary: Three-dimensional electrical impedance tomography (EIT) is considered. Both uniqueness proofs and theoretical reconstruction algorithms available for this problem rely on the use of exponentially growing solutions to the governing conductivity equation. The study of those solutions is continued here. It is shown that exponentially growing solutions exist for low complex frequencies without imposing any regularity assumption on the conductivity. Further, a reconstruction method for conductivities close to a constant is given. In this method the complex frequency is taken to zero instead of infinity. Since this approach involves only moderately oscillatory boundary data, it enables a new class of three-dimensional EIT algorithms, free from the usual high frequency instabilities.

MSC:

35R30 Inverse problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
81U40 Inverse scattering problems in quantum theory
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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