The Calderón problem with partial data in two dimensions

Imanuvilov, Oleg Yu.; Uhlmann, Günther; Yamamoto, Masahiro

doi:10.1090/s0894-0347-10-00656-9

Cited by 141 publications

(196 citation statements)

References 31 publications

(40 reference statements)

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“…It is shown in [87] that for a two dimensional bounded domain 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 one measures the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, one can determine uniquely the conductivity.…”

Section: Partial Data Problem In 2dmentioning

confidence: 99%

“…This implies, for the conductivity equation, that if one measures the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, one can determine uniquely the conductivity. The paper [87] uses Carleman estimates with weights which are harmonic functions with non-degenerate critical points to construct appropriate complex geometrical optics solutions to prove the result. We describe this more precisely below.…”

Section: Partial Data Problem In 2dmentioning

confidence: 99%

“…Denote 0 = ∂ \ . The main result of [87] gives global uniqueness by measuring the Cauchy data on . Let q j ∈ C 2+α ( ), j = 1, 2 for some α > 0 and let q j be complex-valued.…”

Section: Partial Data Problem In 2dmentioning

confidence: 99%

“…In particular we describe briefly the recent work of Astala and Päivärinta proving uniqueness for bounded measurable coefficients, and the work of Bukhgeim proving uniqueness for a potential from Cauchy data associated to the Schrödinger equation. Finally we consider the work of Imanuvilov, Uhlmann and Yamamoto on the partial data problem in two dimensions [87,88]. These sections deal with the case of isotropic conductivities.…”

Section: Introductionmentioning

confidence: 99%

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Inverse problems: seeing the unseen

2014

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This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón's problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?Communicated by Ari Laptev.

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Section: Partial Data Problem In 2dmentioning

confidence: 99%

Section: Partial Data Problem In 2dmentioning

confidence: 99%

“…Denote 0 = ∂ \ . The main result of [87] gives global uniqueness by measuring the Cauchy data on . Let q j ∈ C 2+α ( ), j = 1, 2 for some α > 0 and let q j be complex-valued.…”

Section: Partial Data Problem In 2dmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Inverse problems: seeing the unseen

2014

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“…It is of high practical importance to be able to compute EIT reconstructions from data measured only on a part of the boundary. One possibility for designing such algorithms would be to take one of the recent theoretical breakthroughs, such as [Knu06,KSU07,NS10,IUY10], and implement it in the spirit of (a) above. However, we do not discuss such approaches in this paper.…”

Section: (B)mentioning

confidence: 99%

Nonlinear Inversion from Partial EIT Data:Computational Experiments

Hamilton¹,

Siltanen²

2014

Contemporary Mathematics

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Abstract. Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary, for the necessary small k frequencies.

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