On the inverse boundary value problem for linear isotropic elasticity

Eskin, Gregory; Ralston, James

doi:10.1088/0266-5611/18/3/324

Cited by 93 publications

(94 citation statements)

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“…For the isotropic elasticity system the problem of determining the Lamé parameters from the analog of the DN map in this case which sends the displacement at the boundary to the traction of the boundary has been solved if the Lamé parameter μ is close to a constant [55,142,143].…”

Section: Other Applicationsmentioning

confidence: 99%

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: Other Applicationsmentioning

confidence: 99%

Inverse problems: seeing the unseen

2014

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Section: Other Applicationsmentioning

confidence: 99%

Inverse Problems: Visibility and Invisibility

Uhlmann¹

2013

Journées Équations Aux Dérivées Partielles

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“…Many dedicated mathematical and computational algorithms for the reconstruction of location and parameters of anomalies of different geometrical nature (cavities, cracks, and inclusions) have been proposed over the past few decades (see, for instance, [1,4,6,12,13,14,15,21,26,27,29,31,33,34,42,43,55], the survey articles [16,11], and the monograph [5]). Most of the classical techniques are suited to continuous measurements, in other words, to experimental setups allowing to measure continuum deformations inside the elastic body or on a substantial part of its boundary.…”

Section: Introductionmentioning

confidence: 99%

A Joint Sparse Recovery Framework for Accurate Reconstruction of Inclusions in Elastic Media

Yoo¹,

Jung²,

Lim³

et al. 2017

SIAM J. Imaging Sci.

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A robust algorithm is proposed to reconstruct the spatial support and the Lamé parameters of multiple inclusions in a homogeneous background elastic material using a few measurements of the displacement field over a finite collection of boundary points. The algorithm does not require any linearization or iterative update of Green's function but still allows very accurate reconstruction. The breakthrough comes from a novel interpretation of Lippmann-Schwinger type integral representation of the displacement field in terms of unknown densities having common sparse support on the location of inclusions. Accordingly, the proposed algorithm consists of a two-step approach. First, the localization problem is recast as a joint sparse recovery problem that renders the densities and the inclusion support simultaneously. Then, a noise robust constrained optimization problem is formulated for the reconstruction of elastic parameters. An efficient algorithm is designed for numerical implementation using the Multiple Sparse Bayesian Learning (M-SBL) for joint sparse recovery problem and the Constrained Split Augmented Lagrangian Shrinkage Algorithm (C-SALSA) for the constrained optimization problem. The efficacy of the proposed framework is manifested through extensive numerical simulations. To the best of our knowledge, this is the first algorithm tailored for parameter reconstruction problems in elastic media using highly under-sampled data in the sense of Nyquist rate.

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On the inverse boundary value problem for linear isotropic elasticity

Abstract: We derive three results on the inverse problem of determining the Lamé parameters λ(x) and μ(x) for an isotropic elastic body from its Dirichlet-to-Neumann map.

Cited by 93 publications

References 5 publications

Inverse problems: seeing the unseen

Inverse problems: seeing the unseen

Inverse Problems: Visibility and Invisibility

A Joint Sparse Recovery Framework for Accurate Reconstruction of Inclusions in Elastic Media

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