EIT and the average conductivity

Alessandrini, Giovanni; Cabib, Elio

doi:10.1515/jiip.2008.045

Cited by 11 publications

(20 citation statements)

References 11 publications

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“…Here we just remark that for symmetric conductivity tensors H-convergence reduces to the more usual G-convergence and that M(a, b) and M sym (a, b) are compact with respect to Hconvergence. We also recall that G-or H-convergence has already been shown to be a useful tool in the context of the inverse conductivity problem, see for instance [22,2,16,34,35].…”

Section: 2mentioning

confidence: 99%

A multiscale theory for image registration and nonlinear inverse problems

Modin

Nachman

Rondi

2019

Advances in Mathematics

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In an influential paper, Tadmor, Nezzar and Vese (Multiscale Model. Simul. (2004)) introduced a hierarchical decomposition of an image as a sum of constituents of different scales. Here we construct analogous hierarchical expansions for diffeomorphisms, in the context of image registration, with the sum replaced by composition of maps. We treat this as a special case of a general framework for multiscale decompositions, applicable to a wide range of imaging and nonlinear inverse problems. As a paradigmatic example of the latter, we consider the Calderón inverse conductivity problem. We prove that we can simultaneously perform a numerical reconstruction and a multiscale decomposition of the unknown conductivity, driven by the inverse problem itself. We provide novel convergence proofs which work in the general abstract settings, yet are sharp enough to settle an open problem on the hierarchical decompostion of Tadmor, Nezzar and Vese for arbitrary functions in L 2 . We also give counterexamples that show the optimality of our general results.

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Section: 2mentioning

confidence: 99%

A multiscale theory for image registration and nonlinear inverse problems

Modin

Nachman

Rondi

2019

Advances in Mathematics

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“…The proof of Theorem 1.1 is very different in spirit to that from [1] and we believe it to be of an independent interest. The proof in [1] uses the decay properties of the spherical harmonics away from the boundary. Under some regularity assumptions on σ, which in turn imply certain properties of the corresponding Poisson kernel, a related strategy works (estimating decay properties of solutions with oscillating boundary data away from the boundary).…”

Section: Introductionmentioning

confidence: 97%

G-convergence, Dirichlet to Neumann maps and invisibility

Faraco

Kurylev

Ruiz

2014

Journal of Functional Analysis

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“…This is expected, since extreme oscillations of a sequence of conductivities create an instability of the Calderón problem. This kind of asymptotics is well described via the implication of H or G convergence analysis, [18][19][20] where the homogenization theory assigns the suitable convergence regime. 21,22 As already mentioned, the investigation method depends strongly on the assumptions made for the regularity of the sought conductivity profile.…”

Section: Introductionmentioning

confidence: 99%

The inverse conductivity problem via the calculus of functions of bounded variation

Charalambopoulos

Markaki

Kourounis³

2020

Math Methods in App Sciences

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In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of BV functions. The space of the functions of bounded variation is recommended here as the most appropriate functional space hosting the conductivity profile under reconstruction. For the numerical investigation of the inversion of the inclusion problem, we propose and implement a suitable minimization scheme of an enriched-constructed herein-functional, by exploiting the inner structure of BV space. Finally, we validate and illustrate our theoretical results with numerical experiments.KEYWORDS boundary value problems for second-order elliptic equations, inverse problems MSC CLASSIFICATION 35J25; 35R30Math Meth Appl Sci. 2020;43:5032-5072. wileyonlinelibrary.com/journal/mma

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EIT and the average conductivity

Abstract: We prove the instability of averages of the conductivity in the inverse boundary value\ud problem of Calderòn, also known as the inverse conductivity problem or EIT

Cited by 11 publications

References 11 publications

A multiscale theory for image registration and nonlinear inverse problems

A multiscale theory for image registration and nonlinear inverse problems

G-convergence, Dirichlet to Neumann maps and invisibility

The inverse conductivity problem via the calculus of functions of bounded variation

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