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Electrical Impedance Tomography
 SIAM REVIEW
, 1999
"... This paper surveys some of the work our group has done in electrical impedance tomography. ..."
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Cited by 164 (2 self)
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This paper surveys some of the work our group has done in electrical impedance tomography.
The Calderón problem with partial data
 Ann. of Math. (to
"... In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n≥3, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but ..."
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Cited by 141 (39 self)
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In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n≥3, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem.
An anisotropic inverse boundary value problem
 Comm. Pure Appl. Math
, 1990
"... Abstract. We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ∂Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ∗γ which will produce the same ..."
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Cited by 93 (1 self)
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Abstract. We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ∂Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ∗γ which will produce the same voltage and current measurements on ∂Ω. We prove the converse in two dimensions (i.e. if γ1 and γ2 produce the same boundary measurements, then γ1 = Ψ∗γ2 for an appropriate Ψ) for conductivities which are near a constant. 1 §0. Introduction. The resistance of a wire is defined by Ohm’s law (0.1) δV = IR where δV is the potential difference across the wire and I is the current flow through the wire. Both δV and I are measured quantities and R is defined so that (0.1) holds. The
High contrast impedance tomography
 INVERSE PROBLEMS
, 1996
"... We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The ..."
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Cited by 86 (7 self)
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We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate
Calderón’s inverse problem for anisotropic conductivity
 in the plane, Comm. Partial Differential Equations 30
, 2005
"... Abstract: We study inverse conductivity problem for an anisotropic conductivity σ ∈ L ∞ in bounded and unbounded domains. Also, we give applications of the results in the case when DirichlettoNeumann and NeumanntoDirichlet maps are given only on a part of the boundary. 1. ..."
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Cited by 67 (21 self)
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Abstract: We study inverse conductivity problem for an anisotropic conductivity σ ∈ L ∞ in bounded and unbounded domains. Also, we give applications of the results in the case when DirichlettoNeumann and NeumanntoDirichlet maps are given only on a part of the boundary. 1.
Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction.
, 1998
"... We derive an asymptotic formula for the electrostatic voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective ..."
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Cited by 67 (10 self)
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We derive an asymptotic formula for the electrostatic voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective computational identification procedure. 1. Introduction 2. The electrostatic problem 3. An energy estimate 4. Some additional preliminary estimates 5. An asymptotic formula for the voltage potential 6. Properties of the polarization tensor 7. The continuous dependence of the inhomogeneities 8. Computational results. 9. References 1 Introduction The nondestructive inspection technique known as electrical impedance imaging has recently received considerable attention in the mathematical as well as in the engineering literature [2, 4, 10, 14, 17]. Using this technique one seeks to determine information about the internal conductivity (or impedance) profile of an object based on boundary i...
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
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The Calderón problem with partial data in two dimensions
 J. Amer. Math. Soc
"... Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an ..."
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Cited by 55 (18 self)
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Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results. 1.
Electrical impedance tomography by elastic deformation
 SIAM J. Appl. Math
"... Abstract. This paper presents a new algorithm for conductivity imaging. Our idea is to extract more information about the conductivity distribution from data that have been enriched by coupling impedance electrical measurements to localized elastic perturbations. Using asymptotics of the fields in ..."
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Cited by 54 (13 self)
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Abstract. This paper presents a new algorithm for conductivity imaging. Our idea is to extract more information about the conductivity distribution from data that have been enriched by coupling impedance electrical measurements to localized elastic perturbations. Using asymptotics of the fields in the presence of small volume inclusions, we relate the pointwise values of the energy density to the measured data, through a nonlinear PDE. Our algorithm is based on this PDE and takes full advantage of the enriched data. We give numerical examples that illustrate the performance and the accuracy of our approach. 1.
Electrical impedance tomography and Calderón problem
 INVERSE PROBLEMS
, 2009
"... We survey mathematical developments in the inverse method of Electrical Impedance Tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium. In the mathematical literature this is also known as Calderón’ ..."
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Cited by 52 (1 self)
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We survey mathematical developments in the inverse method of Electrical Impedance Tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium. In the mathematical literature this is also known as Calderón’s problem from Calderón’s pioneer contribution [23]. We concentrate this article around the topic of complex geometrical optics solutions that have led to many advances in the field. In the last section we review some counterexamples to Calderón’s problems that have attracted a lot of interest because of connections with cloaking and invisibility.