Wiener Chaos Expansion and Simulation of Electromagnetic Wave Propagation Excited by a Spatially Incoherent Source

Badieirostami, Majid; Adibi, Ali; Zhou, Haomin; Chow, Shui-Nee

doi:10.1137/090749219

Cited by 18 publications

(28 citation statements)

References 14 publications

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“…In many situations, the source, hence the wave field, may not be deterministic but are rather modeled by random processes [7]. Due to the extra challenge of randomness and uncertainties, little is known for the inverse random source scattering problems.…”

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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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“…The scattering data is obtained by the numerical solution of the stochastic Navier equation instead of the numerical integration of the Fredholm integral equations in order to avoid the so-called inverse crime. Although the stochastic Navier equation may be efficiently solved by using the Wiener Chaos expansions to obtain statistical moments such as the mean and variance [4], we choose the Monte Carlo method to simulate the actual process of measuring data. In each realization, the stochastic Navier equation is solved by using the finite element method with the perfectly matched layer (PML) technique.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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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“…The scattering data is obtained by the numerical solution of the stochastic Helmholtz equation instead of the numerical integration of the Fredholm integral equations in order to avoid the so-called inverse crime. Although the stochastic Helmholtz equation can be more efficiently solved by using the Wiener chaos expansions to obtain statistical moments such as the mean and variance [5], we choose the Monte Carlo method to simulate the actual process of measuring data. In each realization, the stochastic Helmholtz equation is solved by using the finite element method with the perfectly matched layer (PML) technique [13].…”

Section: Numerical Experimentsmentioning

confidence: 99%

Inverse Random Source Scattering Problems in Several Dimensions

Bao¹,

Chen²,

Li³

2016

SIAM/ASA J. Uncertainty Quantification

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This paper concerns the source scattering problems for acoustic wave propagation, which is governed by the two-or three-dimensional stochastic Helmholtz equation. As a source, the electric current density is assumed to be a random function driven by an additive colored noise. Given the random source, the direct problem is to determine the radiated random wave field. The inverse problem is to reconstruct statistical properties of the source from the boundary measurement of the radiated random wave field. In this work, we consider both the direct and inverse problems. We show that the direct problem has a unique mild solution via a constructive proof. Using the mild solution, we derive effective Fredholm integral equations for the inverse problem. A regularized Kaczmarz method is developed by adopting multifrequency scattering data to overcome the challenges of solving the ill-posed and large scale integral equations. Numerical experiments are presented to demonstrate the efficiency of the proposed method. The framework and methodology developed here are expected to be applicable to a wide range of stochastic inverse source problems.

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Wiener Chaos Expansion and Simulation of Electromagnetic Wave Propagation Excited by a Spatially Incoherent Source

Cited by 18 publications

References 14 publications

Inverse Elastic Scattering for a Random Source

Inverse Elastic Scattering for a Random Source

Inverse Random Source Scattering for Elastic Waves

Inverse Random Source Scattering Problems in Several Dimensions

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