Consider the inverse random source scattering problem for the twodimensional timeharmonic elastic wave equation with a linear load. The source is modeled as a microlocally isotropic generalized Gaussian random function whose covariance operator is a classical pseudodifferential operator. The goal is to recover the principal 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 LippmannSchwinger integral equation. For the inverse problem, we demonstrate that, with probability one, the principal 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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Reconstructing a spacedependent source term via the quasireversibility method
The aim of this paper is to solve an important inverse source problem which arises from the wellknown inverse scattering problem. We propose to truncate the Fourier series of the solution to the governing equation with respect to a special basis of L2. By this, we obtain a system of linear elliptic equations. Solutions to this system are the Fourier coefficients of the solution to the governing equation. After computing these Fourier coefficients, we can directly find the desired source function. Numerical examples are presented.
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 Award ID(s):
 2208159
 NSFPAR ID:
 10407410
 Date Published:
 Journal Name:
 Contemporary mathematics
 Volume:
 784
 ISSN:
 27051056
 Page Range / eLocation ID:
 103118
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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