Inverse problems: seeing the unseen

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 tomogra...

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Bibliographic Details
Published inBulletin of mathematical sciences Vol. 4; no. 2; pp. 209 - 279
Main Author Uhlmann, Gunther
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.07.2014
World Scientific Publishing Co. Pte., Ltd
Subjects
Boundary value problems
Electrical impedance
Electrical resistivity
Inverse problems
Mathematics
Mathematics and Statistics
Numerical analysis
Refractivity
Riemann manifold
Tomography
Travel time
Visibility
Brazil
Rio de Janeiro Brazil
Boundary Rigidity Problem
CGO Solutions
Boundary Distance Function
Carleman Estimates
Complex Geometrical Optics (CGO)
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Summary: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. 65–73, 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?
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ISSN:1664-3607
1664-3615
DOI:10.1007/s13373-014-0051-9