Inverse Problems: Visibility and Invisibility

Gunther Uhlmann

Journées Équations aux dérivées partielles (2012)

  • page 1-64
  • ISSN: 0752-0360

Abstract

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This survey article expands on the lectures given at Biarritz in June, 2012, on “Inverse Problems: Visibility and Invisibility". 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 (EIT) and also Calderón’s problem since the famous analyst proposed it in the mathematical literature [38]. 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?

How to cite

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Uhlmann, Gunther. "Inverse Problems: Visibility and Invisibility." Journées Équations aux dérivées partielles (2012): 1-64. <http://eudml.org/doc/275558>.

@article{Uhlmann2012,
abstract = {This survey article expands on the lectures given at Biarritz in June, 2012, on “Inverse Problems: Visibility and Invisibility". 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 (EIT) and also Calderón’s problem since the famous analyst proposed it in the mathematical literature [38]. 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?},
author = {Uhlmann, Gunther},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
pages = {1-64},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Inverse Problems: Visibility and Invisibility},
url = {http://eudml.org/doc/275558},
year = {2012},
}

TY - JOUR
AU - Uhlmann, Gunther
TI - Inverse Problems: Visibility and Invisibility
JO - Journées Équations aux dérivées partielles
PY - 2012
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 64
AB - This survey article expands on the lectures given at Biarritz in June, 2012, on “Inverse Problems: Visibility and Invisibility". 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 (EIT) and also Calderón’s problem since the famous analyst proposed it in the mathematical literature [38]. 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?
LA - eng
UR - http://eudml.org/doc/275558
ER -

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