Highfrequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville
Abstract
We investigate the asymptotic relation between the inverse problems relying on the Helmholtz equation and the radiative transfer equation (RTE) as physical models, in the highfrequency limit. In particular, we evaluate the asymptotic convergence of a generalized version of inverse scattering problem based on the Helmholtz equation, to the inverse scattering problem of the Liouville equation (a simplified version of RTE). The two inverse problems are connected through the Wigner transform that translates the wavetype description on the physical space to the kinetictype description on the phase space, and the Husimi transform that models data localized both in location and direction. The finding suggests that impinging tightly concentrated monochromatic beams can indeed provide stable reconstruction of the medium, asymptotically in the highfrequency regime. This fact stands in contrast with the unstable reconstruction for the classical inverse scattering problem when the probing signals are planewaves.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.03494
 Bibcode:
 2022arXiv220103494C
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematical Physics;
 65N21;
 78A46;
 81S30