Regularity of distributional solutions to stochastic acoustic and elastic scattering problems

Li, Peijun; Wang, Xu

doi:10.1016/j.jde.2021.03.014

Cited by 9 publications

(10 citation statements)

References 12 publications

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“…The random source f satisfies the following assumption. The problem (3.1)-(3.2) was studied in [14,22], where f was assumed to be a R-valued centered GMIG random field of order −m with m ∈ (d − 1, d]. When f is C-valued with its covariance and relation operators being of the same order −m, its regularity is the same as the R-valued case.…”

Section: The Direct Problem Consider the Stochastic Helmholtz Equationmentioning

confidence: 99%

“…When f is C-valued with its covariance and relation operators being of the same order −m, its regularity is the same as the R-valued case. The well-posedness of (3.1)-(3.2) may be obtained directly based on the results in [14,22], but the parameters are not optimal. The following result presents the well-posedness of (3.1)-(3.2), the parameters are different from the existing results and allow more general and rougher sources.…”

Section: The Direct Problem Consider the Stochastic Helmholtz Equationmentioning

confidence: 99%

“…The well-posedness of the problem (6.1)-(6.2) was investigated in [22], where the random source f was assumed to be R d -valued satisfying Assumption 2 with m ∈ (d − 1, d]. The following result gives the well-posedness of the problem (6.1)-(6.2) with a C d -valued random source and a relaxed condition on the order m. Based on the asymptotic behavior of the Green tensor G d given in (2.9), we can rewrite u in (6.3) as the following asymptotic expansion…”

Section: Elastic Wavesmentioning

confidence: 99%

“…As a result, the statistics, such as mean and variance, of the random scatterer are used to quantify the uncertainties of the scatterer and are of more interest in stochastic inverse scattering problems. Recently, stochastic inverse scattering problems have attracted great attention, and many new results are available for various problems, such as random medium problems [22], random potential problems [6,13,[17][18][19], random impedance problems [11], and random surface problems [4,9,12].…”

Section: Introductionmentioning

confidence: 99%

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Inverse source problems for the stochastic wave equations: far-field patterns

Li¹,

Li²,

Wang³

2021

Preprint

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This paper addresses the direct and inverse source problems for the stochastic acoustic, biharmonic, electromagnetic, and elastic wave equations in a unified framework. The driven source is assumed to be a centered generalized microlocally isotropic Gaussian random field, whose covariance and relation operators are classical pseudo-differential operators. Given the random source, the direct problems are shown to be well-posed in the sense of distributions and the regularity of the solutions are given. For the inverse problems, we demonstrate by ergodicity that the principal symbols of the covariance and relation operators can be uniquely determined by a single realization of the far-field pattern averaged over the frequency band with probability one.

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Section: The Direct Problem Consider the Stochastic Helmholtz Equationmentioning

confidence: 99%

Section: The Direct Problem Consider the Stochastic Helmholtz Equationmentioning

confidence: 99%

Section: Elastic Wavesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Inverse source problems for the stochastic wave equations: far-field patterns

Li¹,

Li²,

Wang³

2021

Preprint

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“…The uniqueness was obtained to determine the strength of the potential and source simultaneously based on far-field patterns. Recently, the unique continuation principle was proved in [19] for the second order elliptic operators with rougher potentials or medium parameters of order m ∈ (d − 1, d]. In [16], the rough model was taken to study the inverse random potential problem for the two-dimensional elastic wave equation.…”

Section: Introductionmentioning

confidence: 99%

Inverse scattering for the biharmonic wave equation with a random potential

Li¹,

Wang²

2022

Preprint

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We consider the inverse random potential scattering problem for the two-and threedimensional biharmonic wave equation in lossy media. The potential is assumed to be a microlocally isotropic Gaussian rough field. The main contributions of the work are twofold. First, the unique continuation principle is proved for the fourth order biharmonic wave equation with rough potentials and the well-posedness of the direct scattering problem is established in the distribution sense. Second, the correlation strength of the random potential is shown to be uniquely determined by the high frequency limit of the second moment of the scattered field averaged over the frequency band. Moreover, we demonstrate that the expectation in the data can be removed and the data of a single realization is sufficient for the uniqueness of the inverse problem with probability one when the medium is lossless. PEIJUN LI AND XU WANGThe potential ρ is assumed to be a Gaussian random field defined in a complete probability space (Ω, F, P), where P is the probability measure. More precisely, ρ is required to satisfy the following assumption (cf. [15]).

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Inverse Random Source Scattering for the Helmholtz Equation with Attenuation

Li¹,

Wang²

2021

SIAM J. Appl. Math.

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Regularity of distributional solutions to stochastic acoustic and elastic scattering problems

Cited by 9 publications

References 12 publications

Inverse source problems for the stochastic wave equations: far-field patterns

Inverse source problems for the stochastic wave equations: far-field patterns

Inverse scattering for the biharmonic wave equation with a random potential

Inverse Random Source Scattering for the Helmholtz Equation with Attenuation

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