Lipschitz stability for the inverse conductivity problem

Alessandrini, Giovanni; Vessella, Sergio

doi:10.1016/j.aam.2004.12.002

Cited by 166 publications

(185 citation statements)

References 29 publications

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“…There is the question of whether under some additional a-priori condition one can improve this logarithmic type stability estimate. Alessandrini and Vessella [7] have shown that this is indeed the case and one has a Lipschitz type stability estimate if the conductivity is piecewise constant with jumps on a finite number of domains. Rondi [160] has subsequently shown that the constant in the estimate grows exponentially with the number of domains.…”

Section: Stabilitymentioning

confidence: 88%

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: Stabilitymentioning

confidence: 88%

Inverse problems: seeing the unseen

2014

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“…There is the question of whether under some additional a-priori condition one can improve this logarithmic type stability estimate. Alessandrini and Vessella [7] have shown that this is indeed the case and one has a Lipschitz type stability estimate if the conductivity is piecewise constant with jumps on a finite number of domains. Rondi [161] has subsequently shown that the constant in the estimate grows exponentially with the number of domains.…”

Section: Stabilitymentioning

confidence: 88%

Inverse Problems: Visibility and Invisibility

Uhlmann¹

2013

Journées Équations Aux Dérivées Partielles

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“…One of the main open problems in the stability issue is then to improve this logarithmic-type stability estimate under some additional a priori condition. In [8] it has been shown that (67) can be improved to a Lipschitztype estimate in the case in which is piecewise constant with jumps on a finite number of domains. For piecewise constant complex conductivities a similar result has been proved in [16], where piecewise constant potentials of the Schrödinger equation have been investigated in [18] and Lipschitz stability estimates have been proved in this case as well.…”

Section: Global Stability For Nmentioning

confidence: 97%