Increasing stability for the inverse source scattering problem with multi-frequencies

Li, Peijun; Yuan, Ganghua

doi:10.3934/ipi.2017035

Cited by 59 publications

(34 citation statements)

References 24 publications

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“…It is known that the inverse source problem, in general, does not have a unique solution at a single frequency due to the existence of non-radiating sources [8,17,21,24]. There are two approaches to overcome the issue non-uniqueness: one is to seek the minimum energy solution [33], which represents the pseudo-inverse solution for the inverse source problem; the other is the use of multifrequency data to achieve uniqueness and gain increasing stability [12,14,15,19,30].…”

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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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confidence: 99%

Inverse Elastic Scattering for a Random Source

Li¹,

Li²

2019

SIAM J. Math. Anal.

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“…Since for k ∈ (K, ∞) the data would be unknown, the truncation level of k in I 1 (k), I 2 (k) will keep balance between the value ǫ and the unknown information. The integrands in (13), (14) are analytic functions of ω, hence I 1 (k), I 2 (k) are analytic functions of k in C\(−∞, 0] .…”

Section: Increasing Stability Of the Continuation To High Frequenciesmentioning

confidence: 99%

“…New difficulties in the two-dimensional case are due to the absence of the Huygens principle and a more complicated fundamental solution compared with the three-dimensional case [4]. In [13] the authors studied increasing stability for the source when Ω is a disk.…”

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On increasing stability in the two dimensional inverse source scattering problem with many frequencies

Entekhabi

Isakov

2018

Inverse Problems

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In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain Ω with sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and exact observability for wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate with larger wave numbers interval.

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“…This work is concerned with the inverse source problem of determining the electric current in the time-harmonic Maxwell's system from the multifrequency near-field measurements. For increasing stability analysis concerning the inverse source problem with multi-frequencies, we refer to [10,17,18].…”

Section: Introductionmentioning

confidence: 99%

Solving the multi-frequency electromagnetic inverse source problem by the Fourier method

Wang

Guo

et al. 2018

Journal of Differential Equations

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This work is concerned with an inverse problem of identifying the current source distribution of the time-harmonic Maxwell's equations from multi-frequency measurements. Motivated by the Fourier method for the scalar Helmholtz equation and the polarization vector decomposition, we propose a novel method for determining the source function in the full vector Maxwell's system. Rigorous mathematical justifications of the method are given and numerical examples are provided to demonstrate the feasibility and effectiveness of the method.

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Increasing stability for the inverse source scattering problem with multi-frequencies

Cited by 59 publications

References 24 publications

Inverse Elastic Scattering for a Random Source

Inverse Elastic Scattering for a Random Source

On increasing stability in the two dimensional inverse source scattering problem with many frequencies

Solving the multi-frequency electromagnetic inverse source problem by the Fourier method

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