Limiting Carleman weights and anisotropic inverse problems
TLDR
In this article, the authors considered the anisotropic Calderon problem and related inverse problems, and characterized those Riemannian manifolds which admit limiting Carleman weights, and gave a complex geometrical optics construction for a class of such manifolds.Abstract:
In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.read more
Citations
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Electrical impedance tomography and Calderón's problem
TL;DR: In this paper, the authors 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.
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The Calderón problem with partial data in two dimensions
TL;DR: In this paper, it was shown that the Cauchy data for the Schrodinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential, which is the case for the conductivity equation.
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Inverse problems: seeing the unseen
TL;DR: In this article, the authors deal mainly with two inverse problems and the relation between them, namely voltage and current measurements at the boundary and travel time tomography, which is called electrical impedance tomography and Calderon's problem.
Journal ArticleDOI
Tensor tomography on surfaces
TL;DR: In this paper, it was shown that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective, which solves a long standing inverse problem in the two-dimensional case.
Journal ArticleDOI
The Calderon problem with partial data on manifolds and applications
Carlos E. Kenig,Mikko Salo +1 more
TL;DR: In this article, it was shown that the inverse Calderon problem with partial data can be reduced to the invertibility of a broken geodesic ray transform, where the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction.
References
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Journal ArticleDOI
Calderon's inverse conductivity problem in the plane
Kari Astala,Lassi Päivärinta +1 more
TL;DR: In this paper, it was shown that the Dirichlet to Neumann map for the equation ∇·σ∇u = 0 in a two-dimensional domain uniquely determines the bounded measurable constant.
Journal ArticleDOI
Contróle Exact De Léquation De La Chaleur
G. Lebeau,Luc Robbiano +1 more
TL;DR: In this paper, De La Chaleur et al. present the Controle Exact De Lequation De La CHaleur (CEDE) for Communications in Partial Differential Equations.
Book
Integral Geometry of Tensor Fields
TL;DR: In this article, the authors considered the problem of determining a metric by its hodograph and a linearization of the kinetic equation in a Riemannian manifold, and showed that the ray transform of symmetric tensor fields on Euclidean space can be interpreted as a Fourier transform.
Journal ArticleDOI
The Calderón problem with partial data
TL;DR: In this paper, it was shown that the knowledge of the Cauchy data for the Schr?Nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential.
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