Inverse Random Source Scattering for Elastic Waves

Bao, Gang; Chen, Chuchu; Li, Peijun

doi:10.1137/16m1088922

Cited by 51 publications

(43 citation statements)

References 33 publications

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“…We refer to [17,19,20,24] for the uniqueness and stability theory. Further, several numerical approaches have been developed for shape or source reconstruction [3][4][5]7,10,11,18,[21][22][23][23][24][25]39,45]. For…”

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Inverse elastic scattering problems with phaseless far field data

Liu

2019

Inverse Problems

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This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for inverse elastic scattering problems with phaseless far field data. The phaseless far field data is closely related to the outward energy flux, which is easily measured in practice. Systematically, we study two basic models, i.e., inverse scattering of plane waves by rigid bodies and inverse scattering of sources with compact support. For both models, we show that the phaseless far field data is invariant under translation of the underlying scattering objects, which implies that the location of the objects can not be uniquely recovered by the data. To solve this problem, we consider simultaneously the incident point sources with one fixed source point and at most three scattering strengths. With this technique, we establish some uniqueness results for source scattering problem with multi-frequency phaseless far field data. Furthermore, a fast and stable phase retrieval approach is proposed based on a simple geometric result which provides a stable reconstruction of a point in the plane from three distances to given points. Difficulties arise for inverse scattering by rigid bodies due to the additional unknown far field pattern of the point sources. To overcome this difficulty, we introduce an artificial rigid body into the system and show that the underlying rigid bodies can be uniquely determined by the corresponding phaseless far field data at a fixed frequency. Noting that the far field pattern of the scattered field corresponding to point sources is very small if the source point is far away from the scatterers, we propose an appropriate phase retrieval method for obstacle scattering problems, without using the artificial rigid body. Finally, we propose several sampling methods for shape reconstruction with phaseless far field data directly. For inverse obstacle scattering problems, two different direct sampling methods are proposed with data at a fixed frequency. For inverse source scattering problems, we introduce two direct sampling methods for source supports with sparse multi-frequency data. The phase retrieval techniques are also combined with the classical sampling methods for the shape reconstructions. Extended numerical examples in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed phase retrieval techniques and sampling methods.the readers interested in a more comprehensive treatment of the direct and inverse elastic scattering problems, we suggest consulting [9,14,36,37] on this subject.The two well known difficulties of the inverse scattering problems are nonlinearity and ill-posedness. Actually, in many cases of practical interest, the third difficulty is incomplete data, i.e., only partial information can be measured directly. There are two cases of incomplete data. The first one is the limited-aperture data, where the measurements are only available in limited directions or positions. Limited-aperture data can present a severe ch...

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“…Compute the indicator functional I Θ S (p) for all sampling points p ∈ Z. •(5). Plot the indicator functional I Θ S (p).5.…”

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Inverse elastic scattering problems with phaseless far field data

Liu

2019

Inverse Problems

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“…Due to the extra challenge of randomness and uncertainties, little is known for the inverse random source scattering problems. In [9][10][11]16,27,28], the random source was assumed to be driven by an additive white noise. Mathematical modeling and numerical computation were proposed for a class of inverse source problems for acoustic and elastic waves.…”

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Inverse Elastic Scattering for a Random Source

Li¹,

Li²

2019

SIAM J. Math. Anal.

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Consider the inverse random source scattering problem for the two-dimensional time-harmonic elastic wave equation with an inhomogeneous, anisotropic mass density. The source is modeled as a microlocally isotropic generalized Gaussian random function whose covariance operator is a classical pseudo-differential operator. The goal is to recover the principle symbol of the covariance operator from the displacement measured in a domain away from the source. For such a distributional source, we show that the direct problem has a unique solution by introducing an equivalent Lippmann-Schwinger integral equation. For the inverse problem, we demonstrate that, with probability one, the principle symbol of the covariance operator can be uniquely determined by the amplitude of the displacement averaged over the frequency band, generated by a single realization of the random source. The analysis employs the Born approximation, asymptotic expansions of the Green tensor, and microlocal analysis of the Fourier integral operators.

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“…The corresponding stability analysis was conducted by Li and Yuan in [12,13] with multifrequencies data. Recently Bao et al investigated forward and inverse random source problems for elastic wave equations in [3]. Moreover, an inverse random source problem of the Euler-Bernoulli equation is utilized by Bao et al [5] to determine the unknown material properties, especially for those nanostructures, whose scale are from 1 to 100 nm.…”

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Inverse random source problem for biharmonic equation in two dimensions

Gong¹,

Xu²

2019

Inverse Problems &Amp; Imaging

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The establishment of relevant model and solving an inverse random source problem are one of the main tools for analyzing mechanical properties of elastic materials. In this paper, we study an inverse random source problem for biharmonic equation in two dimension. Under some regularity assumptions on the structure of random source, the well-posedness of the forward problem is established. Moreover, based on the explicit solution of the forward problem, we can solve the corresponding inverse random source problem via two transformed integral equations. Numerical examples are presented to illustrate the validity and effectiveness of the proposed inversion method.

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Inverse Random Source Scattering for Elastic Waves

Cited by 51 publications

References 33 publications

Inverse elastic scattering problems with phaseless far field data

Inverse elastic scattering problems with phaseless far field data

Inverse Elastic Scattering for a Random Source

Inverse random source problem for biharmonic equation in two dimensions

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