High-frequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville
High-frequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville
复制标题
DOI：
10.1137/22m147075x
复制
发表时间：
2022-01
期刊：
影响因子：
--
通讯作者：
Shi Chen;Zhiyan Ding;Qin Li;Leonardo Zepeda-N'unez
中科院分区：
文献类型：
--
作者：
Shi Chen;Zhiyan Ding;Qin Li;Leonardo Zepeda-N'unez
本文研究了以Helmholtz方程和辐射传输方程为物理模型的反问题在高频极限下的渐近关系。特别是，我们评估的Helmholtz方程的逆散射问题的广义版本的基础上，刘维方程（RTE的简化版本）的逆散射问题的渐近收敛。这两个逆问题通过Wigner变换和Husimi变换连接，Wigner变换将物理空间上的波型描述转换为相空间上的动力学型描述，Husimi变换对位置和方向上的数据进行建模。这一发现表明，碰撞紧密集中的单色光束确实可以提供稳定的重建介质，渐近在高频制度。这一事实与经典逆散射问题在探测信号为平面波时的不稳定重建形成了鲜明的对比。
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.