Determining the anisotropic traction state  in a membrane by boundary measurements

Alessandrini, Giovanni; Cabib, Elio

doi:10.3934/ipi.2007.1.437

Cited by 6 publications

(5 citation statements)

References 12 publications

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“…Proposition 6.3 shows that a given σ ∈ Σ(Ω) there exists a unique divergencefree matrix Q such that Λ Q = Λ σ . It has been brought to our attention that proposition 6.3 has also been proven in [5]. Hence, although for a given DtN map there may not exist an isotropic σ such that Λ = Λ σ , there always exists a unique divergence-free Q such that Λ = Λ Q .…”

Section: Electric Impedance Tomographymentioning

confidence: 79%

“…This is of practical importance since the medium to be recovered in a real application may not be isotropic and the associated EIT problem may not admit an isotropic solution. Although the inverse of the map σ → Λ σ is not continuous with respect to the topology of G-convergence when σ is restricted to the set of isotropic matrices, it has also been shown in section 3 of [5] that this inverse is continuous with respect to the topology of G-convergence when σ is restricted to the set of divergence-free matrices.…”

Section: Electric Impedance Tomographymentioning

confidence: 99%

“…We show that the EIT problem admits a unique solution in the space of divergence-free matrices. The uniqueness property has also been obtained in [5]. When conductivities are endowed with the topology of G-convergence the inverse conductivity problem is discontinuous when restricted to isotropic matrices [5,53] and continuous when restricted to divergence-free matrices [5].…”

Section: Introductionmentioning

confidence: 99%

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Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization

Desbrun

Donaldson

Owhadi

2013

Multiscale Analysis and Nonlinear Dynamics

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Imaging and simulation methods are typically constrained to resolutions much coarser than the scale of physical micro-structures present in body tissues or geological features. Mathematical and numerical homogenization address this practical issue by identifying and computing appropriate spatial averages that result in accuracy and consistency between the macro-scales we observe and the underlying micro-scale models we assume. Among the various applications benefiting from homogenization, Electric Impedance Tomography (EIT) images the electrical conductivity of a body by measuring electrical potentials consequential to electric currents applied to the exterior of the body. EIT is routinely used in breast cancer detection and cardio-pulmonary imaging, where current flow in fine-scale tissues underlies the resulting coarse-scale images.In this paper, we introduce a geometric approach for the homogenization (simulation) and inverse homogenization (imaging) of divergenceform elliptic operators with rough conductivity coefficients in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions over the domain. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of the domain when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. We explicitly give the transformations which map conductivity coefficients into divergencefree matrices and convex functions, as well as their respective inverses. Using weighted Delaunay triangulations for linearly interpolating convex functions, we apply this geometric framework to obtain a robust homogenization algorithm for arbitrary rough coefficients, extending the global optimality of Delaunay triangulations with respect to a discrete Dirichlet energy to weighted Delaunay triangulations. We then consider inverse homogenization, which is known to be both non-linear and severely illposed, but that we can decompose into a linear ill-posed problem and a well-posed non-linear problem. Finally, our new geometric approach to homogenization and inverse homogenization is applied to EIT.

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Section: Electric Impedance Tomographymentioning

confidence: 79%

Section: Electric Impedance Tomographymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization

Desbrun

Donaldson

Owhadi

2013

Multiscale Analysis and Nonlinear Dynamics

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show abstract

“…Stability with respect to the G-convergence has been proved in 2D by Alessandrini and Cabib in [2] assuming further that ∇ • σ = 0. As discussed also in that paper, the lack of uniqueness in the Calderón problem in the anisotropic case prevents stability in the general case.…”

Section: Stability With Respect To the G-convergencementioning

confidence: 99%

“…Recall that if F is a diffeomorphism of Ω, which is the identity at the boundary, then Λ F * (σ) = Λ σ . As discussed for example in [2], the isotropic conductivities are G-dense in the set of anisotropic conductivities, so that the only hope is to recover from the D-N maps the G-limit up to a gauge transformation. In contrast to the previous results on the conditional stability, the compactness of the sets M K in the G-topology indeed provides a stability result which is unconditional respect to regularity (we still require ellipticity).…”

Section: Introductionmentioning

confidence: 99%