AI 中文总结
研究从弹性散射远场测量重建不可穿透散射体形状位置的反问题，提出基于单调性的形状表征定理及算法，包括基于计数的采样法和利用负特征值大小的谱采样算法，数值实验验证了算法在不同条件下对复杂形状的重建效果。
AI 中文摘要
从远场测量重建未知不可穿透散射体的位置和形状是弹性散射中的一个基本反问题。本文针对刚性和无牵引不可穿透散射体提出了基于单调性的形状表征定理并开发了相应算法。通过建立弹性远场算子的因式分解并构造局部波函数，得出用于确定不可穿透散射体形状和位置的基于单调性的尖锐表征准则。在此理论基础上，先提出基于计数的单调性采样方法，为解决特征值计数对测量噪声的固有敏感性，又开发了利用负特征值大小的两种新型单调性谱采样算法。单频算法对数据扰动具有稳健稳定性，多频算法将频率信息聚合为多尺度指标，兼顾噪声鲁棒性和高分辨率几何保真度。各种散射体几何形状和噪声水平的数值实验证明了两种算法的有效性。
英文摘要
Reconstructing the location and shape of an unknown impenetrable scatterer from far-field measurements is a fundamental inverse problem in elastic scattering. In this paper, we propose monotonicity-based shape characterization theorems and develop corresponding algorithms for rigid and traction-free impenetrable scatterers. By establishing the factorization of the elastic far-field operator and constructing localized wave functions, we derive a sharp monotonicity-based characterization criterion for determining the shape and position of the impenetrable scatterer. This criterion is based on the spectral properties of the \emph{monotonicity operator}, defined as a specific linear combination of the far-field and Herglotz probing operators. Building on this theoretical foundation, we first present a counting-based monotonicity sampling method that evaluates the number of negative eigenvalues of the monotonicity operator. To address the inherent sensitivity of eigenvalue-counting to measurement noise, we further develop two novel monotonicity spectral sampling algorithms that exploit the magnitudes, rather than merely the signs, of the negative eigenvalues. The single-frequency monotonicity spectral sampling method provides robust stability against data perturbations, while the multi-frequency monotonicity spectral sampling method extension aggregates frequency information into a multiscale indicator that balances noise robustness with high-resolution geometric fidelity. Numerical experiments across various scatterer geometries and noise levels demonstrate sharp boundary localization and accurate reconstruction of complex concave features, confirming the effectiveness of the single-frequency and multi-frequency monotonicity spectral sampling methods.
