A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements

Audibert, Lorenzo; Haddar, Houssem

doi:10.1088/0266-5611/30/3/035011

Cited by 70 publications

(169 citation statements)

References 17 publications

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“…0 is the first kind Hankel function of order zero and u is the total field for (1)- (2). We consider the boundary of the domain ∂D = r(θ) (cos(θ), sin(θ)) where r(θ) is given by See Figure 1 for a plot of the pear-shaped, star-shaped and peanut-shaped domain.…”

Section: Numerical Results In Two Dimensionsmentioning

confidence: 99%

Analysis of new direct sampling indicators for far-field measurements

Harris

Kleefeld

2019

Inverse Problems

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This article focuses on the analysis of three direct sampling indicators which can be used for recovering scatterers from the far-field pattern of time-harmonic acoustic measurements. These methods fall under the category of sampling methods where an indicator function is constructed using the far-field operator. Motivated by some recent work, we study the standard indicator using the far-field operator and two indicators derived from the factorization method. We show equivalence of two indicators previously studied as well as propose a new indicator based on the Tikhonov regularization applied to the far-field equation for the factorization method. Finally, we give some numerical examples to show how the reconstructions compare to other direct sampling methods.

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Section: Numerical Results In Two Dimensionsmentioning

confidence: 99%

Analysis of new direct sampling indicators for far-field measurements

Harris

Kleefeld

2019

Inverse Problems

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“…The approach used in this paper provides a criteria to reconstruct the support of anomalies without explicitly know or reconstruct the background. The imaging method is based on the generalized arXiv:1807.11828v1 [math-ph] 31 Jul 2018 linear sampling method which was first introduced in [3], [5]. This method falls in the class of qualitative approaches to inverse scattering.…”

Section: Introductionmentioning

confidence: 99%

New interior transmission problem applied to a single Floquet–Bloch mode imaging of local perturbations in periodic media

Cakoni¹,

Haddar²,

Nguyen³

2018

Inverse Problems

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This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use a differential linear sampling method to reconstruct the support of perturbations without using the Green's function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior transmission problem, which until now was an open problem except for the special case when the local perturbation didn't intersect the background inhomogeneities. The analysis of this new interior transmission problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.

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“…is the fourth-order elasticity tensor; I m (m = 2, 4) denotes the mth-order symmetric identity tensor; t f = n · C : ∇u f is the free-field traction vector; n = n − is the unit normal on Γ, and L : 3 represents a heterogeneous bijective contact law over the fracture surface, physically relating the displacement jump to surface traction. In many practical situations, the fracture's contact law is linearized about a dynamic equilibrium state as…”

Section: Problem Statementmentioning

confidence: 99%

“…the illumination frequency. This is pursued by drawing from the theories of inverse scattering [12,14] and, in particular, by building upon the Factorization Method [21,8] and the recently developed Generalized Linear Sampling Method (GLSM) [2,3] which completes the theoretical foundation of its LSM predecessor. First, the inverse problem is formulated in the frequency domain where the illuminating wavefield is described by the elastic Herglotz wave function [16] with its inherent compressional (P) and shear (S) wave components.…”

Section: Introductionmentioning

confidence: 99%

Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures

2017

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Abstract. A theoretical foundation is developed for active seismic reconstruction of fractures endowed with spatially-varying interfacial condition (e.g. partially-closed fractures, hydraulic fractures). The proposed indicator functional carries a superior localization property with no significant sensitivity to the fracture's contact condition, measurement errors, and illumination frequency. This is accomplished through the paradigm of the F -factorization technique and the recently developed Generalized Linear Sampling Method (GLSM) applied to elastodynamics. The direct scattering problem is formulated in the frequency domain where the fracture surface is illuminated by a set of incident plane waves, while monitoring the induced scattered field in the form of (elastic) far-field patterns. The analysis of the wellposedness of the forward problem leads to an admissibility condition on the fracture's (linearized) contact parameters. This in turn contributes toward establishing the applicability of the F -factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such minimizer is then used to construct a robust fracture indicator function, whose performance is illustrated through a set of numerical experiments. For completeness, the results of the GLSM reconstruction are compared to those obtained by the classical linear sampling method (LSM).

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A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements

Cited by 70 publications

References 17 publications

Analysis of new direct sampling indicators for far-field measurements

Analysis of new direct sampling indicators for far-field measurements

New interior transmission problem applied to a single Floquet–Bloch mode imaging of local perturbations in periodic media

Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures

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