High-frequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville

TL;DR

This paper explores how inverse scattering problems based on the Helmholtz equation asymptotically converge to those based on the Liouville equation in the high-frequency limit, highlighting stable reconstruction possibilities.

Contribution

It demonstrates the asymptotic convergence of inverse Helmholtz problems to inverse Liouville problems using Wigner and Husimi transforms, revealing stable high-frequency reconstruction methods.

Findings

Tightly concentrated monochromatic beams enable stable medium reconstruction at high frequencies.

Classical plane-wave probing leads to unstable inverse scattering reconstructions.

The convergence links wave-based and kinetic-based inverse problems asymptotically.

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 high-frequency 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 wave-type description on the physical space to the kinetic-type 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 high-frequency regime. This fact stands in contrast with the unstable reconstruction for the classical inverse scattering problem when the probing signals are plane-waves.