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反问题与不确定性量化研讨会 (Workshop on Inverse Problems and Uncertainty Quantification)

Inverse Elastic Scattering for a Random Potential

Speaker

李建樑 , 湖南师范大学

Time

18 Nov, 10:40 - 11:10

Abstract

This talk is concerned with an inverse scattering problem for the time-harmonic elastic wave equation with a random potential. Interpreted as a distribution, the potential is assumed to be a micro locally isotropic generalized Gaussian random field with the covariance operator being described by a classical pseudo-differential operator. The goal is to determine the principal symbol of the covariance operator from the scattered wave measured in a bounded domain which has a positive distance from the domain of the potential. For such a rough potential, the well-posedness of the direct scattering problem in the distribution sense is established by studying an equivalent Lippmann–Schwinger integral equation. For the inverse scattering problem, it is shown with probability one that the principal symbol of the covariance operator can be uniquely determined by the amplitude of the scattered waves averaged over the frequency band from a single realization of the random potential. The analysis employs the Born approximation in high frequency, asymptotics of the Green tensor for the elastic wave equation, and microlocal analysis for the Fourier integral operators.