Reconstruction of Coefficients in Scalar Second‐Order Elliptic Equations from Knowledge of Their Solutions

Bal, Guillaume; Uhlmann, Günther

doi:10.1002/cpa.21453

Cited by 87 publications

(119 citation statements)

References 28 publications

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“…In this manner, the above inverse source problem is converted into a so-called hybrid inverse problem, where an external field is used to control the material properties of a medium of interest, which is then probed by a second field. See [3,4,5,6,7,8,9,10,13,19,21,22,24,25,26,27] for examples of hybrid inverse problems in other physical settings.…”

Section: Introductionmentioning

confidence: 99%

Ultrasound Modulated Bioluminescence Tomography and Controllability of the Radiative Transport Equation

Bal¹,

Chung²,

Schotland³

2016

SIAM J. Math. Anal.

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Section: Introductionmentioning

confidence: 99%

Ultrasound Modulated Bioluminescence Tomography and Controllability of the Radiative Transport Equation

Bal¹,

Chung²,

Schotland³

2016

SIAM J. Math. Anal.

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“…Included here for comparison, the reconstruction µ (2) of the shear modulus in Figs. 3b and 5b, and that is incorrectly based on only one deviatoric strain field measurement, yields larger discrepancies with the reference compared to µ (2,3) . The latter is indeed computed using two strain field measurements as required from Proposition 2.…”

Section: Numerical Examplesmentioning

confidence: 98%

Converting strain maps into elasticity maps for materials with small contrast

Bellis

2017

Journal of the Mechanics and Physics of Solids

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This study addresses the question of the quantitative reconstruction of heterogeneous distributions of isotropic elastic moduli from full strain field data. This parameter identification problem exposes the need for a local reconstruction procedure that is investigated here in the case of materials with small contrast. To begin with the integral formulation framework for the periodic linear elasticity problem, first-and second-order asymptotics are retained for the strain field solution and the effective elasticity tensor. Properties of the featured Green's tensor are investigated to characterize its decomposition into an isotropic term and an orthogonal part. The former is then shown to define a local contribution to the volume integral equations considered. Based on this property, then the combination of multiple strain field solutions corresponding to well-chosen applied macroscopic strains is shown to lead to a set of local and uncoupled identities relating, respectively, the bulk and shear moduli to the spherical and deviatoric components of the strain fields. Valid at the first-order in the weak contrast limit, such relations permit point-wise conversions of strain maps into elasticity maps. Furthermore, it is also shown that for macroscopically isotropic material configurations a single strain field solution is actually sufficient to reconstruct either the bulk or the shear modulus distribution. Those results are then revisited in the case of bounded media. Finally, some sets of analytical and numerical examples are provided for comparison and to illustrate the relevance of the obtained strain-modulus local equations for a parameter identification method based on full-field data.

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“…See [13,176] for a review of some of these inverse methods. CGO solutions have been used in these hybrid methods in [15][16][17][18]42,110]. Inverse transport see [14] for applications of 60 solutions to inverse problem for the transport equation.…”

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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Reconstruction of Coefficients in Scalar Second‐Order Elliptic Equations from Knowledge of Their Solutions

Cited by 87 publications

References 28 publications

Ultrasound Modulated Bioluminescence Tomography and Controllability of the Radiative Transport Equation

Ultrasound Modulated Bioluminescence Tomography and Controllability of the Radiative Transport Equation

Converting strain maps into elasticity maps for materials with small contrast

Inverse problems: seeing the unseen

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