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252
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 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.
2007 On uniqueness in the inverse conductivity problem with local data Inverse Probl
 Imaging
"... The inverse condictivity problem with many boundary measurements consists of recovery of conductivity coefficient a (principal part) of an elliptic equation in a domain Ω ⊂ Rn, n = 2, 3 from the Neumann data given for all Dirichlet data (DirichlettoNeumann map). Calderon [5] proposed the ..."
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Cited by 44 (0 self)
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The inverse condictivity problem with many boundary measurements consists of recovery of conductivity coefficient a (principal part) of an elliptic equation in a domain Ω ⊂ Rn, n = 2, 3 from the Neumann data given for all Dirichlet data (DirichlettoNeumann map). Calderon [5] proposed the
An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem. Inverse Problems
, 2000
"... Abstract. The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ⊂ R2 from knowledge of the DirichlettoNeumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) appli ..."
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Cited by 43 (12 self)
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Abstract. The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ⊂ R2 from knowledge of the DirichlettoNeumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) applications. In 1996 A Nachman proved that the DirichlettoNeumann map uniquely determines C2 conductivities. The proof, which is constructive, outlines a direct method for reconstructing the conductivity. In this paper we present an implementation of the algorithm in Nachman’s proof. The paper includes numerical results obtained by applying the general algorithms described to two radially symmetric cases of small and large contrast. (Some figures in this article are in colour only in the electronic version; see www.iop.org) 1.
A global stability estimate for the Gel’fandCalderón inverse problem in two dimensions
, 2010
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Invisibility and Inverse Problems
, 2008
"... We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a var ..."
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Cited by 32 (19 self)
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We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a variety of wave phenomena, including electrostatics, electromagnetism, acoustics and quantum mechanics, have transformation laws under changes of variables which allow one to design material parameters that steer waves around a hidden region, returning them to their original path on the far side. Not only are observers unaware of the contents of the hidden region, they are not even aware that something is being hidden; the object, which casts no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other devices having striking effects on wave propagation, unseen in nature, are also possible. These designs are initially based on the transformation laws of the relevant PDEs, but due to the singular transformations needed for the desired effects, care needs to be taken in formulating and analyzing physically meaningful solutions. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved.
The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction
 Comm. Pure Appl. Math
"... The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R n. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than th ..."
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Cited by 31 (21 self)
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The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R n. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than that of the delta function
Formulae and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential
 Inverse Problems
"... For the Schrödinger equation at fixed energy with a potential supported in a bounded domain we give formulas and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential. For the case of zero background potential these results were obtained in [R.G.Nov ..."
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Cited by 31 (13 self)
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For the Schrödinger equation at fixed energy with a potential supported in a bounded domain we give formulas and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential. For the case of zero background potential these results were obtained in [R.G.Novikov, Multidimensional inverse spectral problem for the equation −∆ψ + (v(x) − Eu(x))ψ = 0, Funkt. Anal. i Ego Prilozhen
Cloaking Devices, Electromagnetic Wormholes and Transformation Optics
"... We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves. Ideas for devices that would have once seemed fanciful may now be at least approximately implemented physically using a new class of artificially structured materials called m ..."
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Cited by 30 (7 self)
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We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves. Ideas for devices that would have once seemed fanciful may now be at least approximately implemented physically using a new class of artificially structured materials called metamaterials. Maxwell’s equations have transformation laws that allow for design of electromagnetic material parameters that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other designs having striking effects on wave propagation are possible. All of these designs are initially based on the transformation laws of the equations that govern wave propagation but, due