论文标题
反向散射问题的高频极限：从反赫尔姆霍尔茨到倒物的渐近收敛性
High-frequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville
论文作者
论文摘要
我们将依赖于Helmholtz方程和辐射转移方程（RTE）作为物理模型的逆问题之间的渐近关系在高频限制下。特别是，我们将基于Helmholtz方程的广义反向散射问题的渐近收敛性评估为liouville方程的逆散射问题（RTE的简化版本）。这两个反问题是通过Wigner变换连接的，该变换将物理空间上的波型描述转化为相位空间上的动力学类型描述，而Husimi变换模拟了在位置和方向上既定位置的数据进行模型。该发现表明，撞击紧密浓缩的单色束确实可以在高频制度中渐近地提供介质的稳定重建。当探测信号是平面波时，这一事实与经典反向散射问题的不稳定重建相反。
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.
