The Inverse Conductivity Problem with One Measurement: Stability and Estimation of Size

Abstract. In this paper we provide a framework for constructing general complex geometrical optics solutions for several systems of two variables that can be reduced to a system with the Laplacian as the leading order term. We apply these special solutions to the problem of reconstructing inclusions inside a domain filled with known conductivity from local boundary measurements. Computational results demonstrate the versatility of these solutions to determine electrical inclusions. Key words. geometrical optics, inclusions, numerical algorithm AMS subject classifications. 35R30, 65N21DOI. 10.1137/0606763501. Introduction. Inverse boundary value problems are a class of inverse problems where one attempts to determine the internal parameters of body by making measurements only at the surface of the body. A prototypical example that has received a lot of attention is electrical impedance tomography (EIT). In this inverse method one would like to determine the conductivity distribution inside a body by making voltage and current measurements at the boundary.There are many applications of EIT ranging from early breast cancer detection [32] to geophysical sensing for underground objects; see [18], [24], [25], [27]. The article [28] and the ones reviewed in [29] assume that the measurements are made on the whole boundary. However, it is often possible to make the measurements only on part of the boundary; this is the partial data problem. This is the case for the applications in breast cancer detection and geophysical sensing mentioned above.The boundary information is encoded into the Dirichlet-to-Neumann map associated with the conductivity equation. More precisely, let Ω be an open bounded domain with smooth boundary in R d with d = 2 or 3. Assume that γ(x) > 0 in Ω possesses a suitable regularity. The conductivity equation is described by the following elliptic equation:

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“…To describe our method, we begin with the following integral inequalities given in [19] (also see [10] for a proof).…”

Section: Inverse Problemsmentioning

confidence: 99%

Reconstructing Discontinuities Using Complex Geometrical Optics Solutions

Uhlmann

Wang²

2008

SIAM J. Appl. Math.

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“…In particular, since such a power gap contains information at the accessible boundary, it is possible to extend this to the inaccessible part of the boundary in a quantitative manner and thus obtain information on its size. This procedure follows the lines of similar problems studied in [3,15] and later developed in [5, 8, 10-12, 19, 20]. The main novelty of this paper relies on the evaluation of a defect located on the boundary.…”

Section: Introductionmentioning

confidence: 96%

“…More precisely, the approach we follow is that proposed by Alessandrini and Rosset [3] and Kang et al . [15] and subsequently refined by Alessandrini et al . [4].…”

Section: Introductionmentioning

confidence: 99%

Size estimates of unknown boundaries with a Robin-type condition

Cristo¹,

Sincich²,

Vessella³

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…This is a well-known fact in the theory of EIT and has been used in many instances in the past [2,3,11,14,4].…”

Section: Introductionmentioning

confidence: 99%

Depth dependent resolution in Electrical Impedance Tomography

Alessandrini

Scapin

2017

Journal of Inverse and Ill-Posed Problems

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Abstract. We consider the two-dimensional version of Calderòn's problem. When the D-N map is assumed to be known up to an error level ε 0 , we investigate how the resolution in the determination of the unknown conductivity deteriorates the farther one goes from the boundary. We provide explicit formulas for the resolution, which apply to conductivities which are perturbations, concentrated near an interior point q, of the homogeneous conductivity.

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The Inverse Conductivity Problem with One Measurement: Stability and Estimation of Size

Cited by 102 publications

References 12 publications

Reconstructing Discontinuities Using Complex Geometrical Optics Solutions

Reconstructing Discontinuities Using Complex Geometrical Optics Solutions

Size estimates of unknown boundaries with a Robin-type condition

Depth dependent resolution in Electrical Impedance Tomography

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