衍射波的逆散射
Inverse Scattering by Diffracted Waves
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中文总结 AI 辅助
研究利用衍射波检测刚性拐角障碍物，通过镜像法转化问题，借助Cheeger-Taylor泛函演算和微局部分析获取信息，证明给定脉冲下衍射波测量值可确定障碍物可见部分位置和形状，且接收器灵活，无需求解拐角域波动方程。
中文摘要 AI 辅助
除反射和折射外，波的另一种偏差形式是衍射。当入射波撞击角落时，衍射波从角落尖端发出并全向传播。本文提出了一个利用测量的衍射波数据检测刚性拐角障碍物的新框架。该框架首先通过镜像法将初始边值问题转化为圆锥流形上的初值问题。随后，通过Cheeger-Taylor泛函演算和圆锥流形上的微局部分析来获取障碍物信息。具体而言，证明了对于给定脉冲，曲线接收器捕获的衍射波测量值能唯一确定多边形障碍物可见部分的位置和形状，且证明是建设性的，明确制定了相应的恢复方案。该方法有两个关键优势：一是接收器的大小和位置可以任意；二是反演仅需衍射波测量值，无需像传统方法那样求解拐角域内的波动方程。
英文摘要
In addition to reflection and refraction, another form of wave deviation is defined as diffraction. Notably, when incident waves strike a corner, diffracted waves emanate from the corner tip and propagate omnidirectionally. This paper proposes a novel framework for detecting rigid cornered obstacles using measured diffracted wave data. The framework first transforms the underlying initial-boundary value problems into initial value problems on conic manifolds via the method of images. Subsequently, the retrieval of obstacle information is achieved through Cheeger--Taylor functional calculus and microlocal analysis on conic manifolds. Specifically, we prove that for a given pulse, measurements of the resulting diffracted waves captured by a curve receiver uniquely determine both the location and shape of the visible portion of a polygonal obstacle. The proof is constructive, explicitly formulating the corresponding recovery scheme. This methodology offers two key advantages: first, the size and placement of the receiver can be arbitrary; second, the inversion only requires measurements of diffracted waves and obviates the need to solve wave equations within the cornered domain as in conventional methods.
