Recent progress in the Calderón problem with partial data

Kenig, Carlos E.; Salo, Mikko

doi:10.1090/conm/615/12245

Cited by 83 publications

(75 citation statements)

References 61 publications

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“…Recent results are also provided by Kenig and Salo [71,72]. It is worth noting that [72] generalizes the results obtained in both [73] and [68] by making use of improved Carleman estimates with boundary terms, CGO solutions involving reflected Gaussian beam quasimodes and invertibility of (broken) geodesics ray transforms.…”

Section: Some Remarks On the Dirichlet-to-neumann Map Eit With Partiasupporting

confidence: 55%

Electrical Impedance Tomography

Adler¹,

Gaburro²,

Lionheart³

2015

Handbook of Mathematical Methods in Imaging

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Section: Some Remarks On the Dirichlet-to-neumann Map Eit With Partiasupporting

confidence: 55%

Electrical Impedance Tomography

Adler¹,

Gaburro²,

Lionheart³

2015

Handbook of Mathematical Methods in Imaging

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“…These results have been further extended to the case of partial data in [22]. We refer to [13,16,17] for additional results in the case of local data and to the surveys [14,23] for further references.…”

Section: Does This Imply Thatmentioning

confidence: 90%

Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets

Daudé

Kamran²,

Nicoleau³

2019

Annales De l'Institut Fourier

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After giving a general introduction to the main known results on the anisotropic Calderón problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5,6] for the anisotropic Calderón problem at fixed frequency, in dimension n ≥ 3, when the Dirichlet and Neumann data are measured on disjoint subsets of the boundary. These non-uniqueness results are of the following nature: given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension n ≥ 3, we first show that there exist in the conformal class of g an infinite number of Riemannian metricsg such that their corresponding Dirichlet-to-Neumann maps at a fixed frequency coincide when the Dirichlet data ΓD and Neumann data ΓN are measured on disjoint sets and satisfy ΓD ∪ ΓN = ∂M . The corresponding conformal factors satisfy a nonlinear elliptic PDE of Yamabe type on (M, g) and arise from a natural but subtle gauge invariance of the Calderón when the data are given on disjoint sets. We then present counterexamples to uniqueness in dimension n ≥ 3 to the anisotropic Calderón problem at fixed frequency with data on disjoint sets, which do not arise from this gauge invariance. They are given by cylindrical Riemannian manifolds with boundary having two ends, equipped with a suitably chosen warped product metric. This survey concludes with some remarks on the case of manifolds with corners.

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“…In the CEM, we measure The question whether full or local Neumann-Dirichlet-measurements uniquely determine the coefficient function σ has become famous under the name Calderón problem [23,24], and has been intensively studied in the mathematical literature due to its practical relevance for EIT and many other related inverse coefficient problems, cf. [66,67,28,83,76,7,9,58,63,54,35,34,61,62,55,25,68].…”

Section: Introductionmentioning

confidence: 99%

Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes

Harrach

2019

Inverse Problems

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For the linearized reconstruction problem in Electrical Impedance Tomography (EIT) with the Complete Electrode Model (CEM), Lechleiter and Rieder (2008 Inverse Problems 24 065009) have shown that a piecewise polynomial conductivity on a fixed partition is uniquely determined if enough electrodes are being used. We extend their result to the full non-linear case and show that measurements on a sufficiently high number of electrodes uniquely determine a conductivity in any finite-dimensional subset of piecewise-analytic functions. We also prove Lipschitz stability, and derive analogue results for the continuum model, where finitely many measurements determine a finite-dimensional Galerkin projection of the Neumann-to-Dirichlet operator on a boundary part.

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Recent progress in the Calderón problem with partial data

Abstract: We survey recent results on Calderón's inverse problem with partial data, focusing on three and higher dimensions.

Cited by 83 publications

References 61 publications

Electrical Impedance Tomography

Electrical Impedance Tomography

Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets

Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes

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