Inverse Random Source Scattering Problems in Several Dimensions

Bao, Gang; Chen, Chuchu; Li, Peijun

doi:10.1137/16m1067470

Cited by 57 publications

(54 citation statements)

References 31 publications

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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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confidence: 99%

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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Inverse Elastic Scattering for a Random Source

Li¹,

Li²

2019

SIAM J. Math. Anal.

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“…The determination of a random source by the corresponding passive measurement was also recently studied in [3,24,30], and the determination of a random potential by the corresponding active measurement was established in [5]. We also refer to [19] and the references therein for more relevant studies on the determination of a random potential.…”

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confidence: 99%

Determining a Random Schrödinger Equation with Unknown Source and Potential

Liu²,

2019

SIAM J. Math. Anal.

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We are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrödinger equation with unknown source and potential terms. The well-posedness of the direct scattering problem is first established. Three uniqueness results are then obtained for the corresponding inverse problems in determining the variance of the source, the potential and the expectation of the source, respectively, by the associated far-field measurements. First, a single realization of the passive scattering measurement can uniquely recover the variance of the source without the a priori knowledge of the other unknowns. Second, if active scattering measurement can be further obtained, a single realization can uniquely recover the potential function without knowing the source. Finally, both the potential and the first two statistic moments of the random source can be uniquely recovered with full measurement data. The major novelty of our study is that on the one hand, both the random source and the potential are unknown, and on the other hand, both passive and active scattering measurements are used for the recovery in different scenarios.

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“…Recently, one-dimensional stochastic inverse source problems have been considered in [6,10,27], where the governing equations are stochastic ordinary differential equations. Utilizing the Green functions, the authors have presented the first approach in [5] for solving the inverse random source scattering problem in higher dimensions, where the stochastic partial differential equations are considered.…”

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“…This work is a nontrivial extension of the method proposed in [5] for the inverse random source scattering problem of the stochastic Helmholtz equation, to solve the inverse random source scattering problem of the stochastic Navier equation. Clearly, the elastic wave equation is more challenging due to the coexistence of compressional waves and shear waves that propagate at different speeds.…”

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Inverse Random Source Scattering for Elastic Waves

Bao¹,

Chen²,

Li³

2017

SIAM J. Numer. Anal.

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Abstract. This paper is concerned with the direct and inverse random source scattering problems for elastic waves where the source is assumed to be driven by an additive white noise. Given the source, the direct problem is to determine the displacement of the random wave field. The inverse problem is to reconstruct the mean and variance of the random source from the boundary measurement of the wave field at multiple frequencies. The direct problem is shown to have a unique mild solution by using a constructive proof. Based on the explicit mild solution, Fredholm integral equations of the first kind are deduced for the inverse problem. The regularized Kaczmarz method is presented to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

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Inverse Random Source Scattering Problems in Several Dimensions

Cited by 57 publications

References 31 publications

Inverse Elastic Scattering for a Random Source

Inverse Elastic Scattering for a Random Source

Determining a Random Schrödinger Equation with Unknown Source and Potential

Inverse Random Source Scattering for Elastic Waves

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