发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对二维双调和Helmholtz方程，提出一种通过远场变换实现近场测量数据因子分解的夹紧障碍物重建方法，并验证了其有效性。
AI 中文摘要
本文考虑了一个逆形状问题，即从频域中双调和Helmholtz方程的点源和偶极子近场测量数据中恢复二维空间中一个未知的不可穿透的夹紧障碍物。测量数据包括散射场及其在围绕障碍物的闭合测量曲线上的法向导数。由于相关的近场算子不直接允许因子分解方法所需的对称分解，我们引入了一个远场变换。该变换独立于障碍物定义，并将近场算子扩展为相关的远场算子，该远场算子确实允许对称分解。这通过因子分解方法理论给出了障碍物的严格完整表征，并基于变换算子的谱数据产生了一个实用的重建算法。数值实验证明了所提方法的有效性，其中合成近场数据使用基本解方法生成。我们还考虑了仅使用点源生成的散射场测量进行重建的情况，展示了减少测量数据量的潜力。
英文摘要
This paper considers an inverse shape problem for recovering an unknown impenetrable clamped obstacle in two dimensions from near-field point source and dipole measurements for the biharmonic Helmholtz equation in the frequency domain. The measured data consist of the scattered field and its normal derivative on a closed measurement curve surrounding the obstacle. Since the associated near-field operator does not directly admit the symmetric factorization required by the factorization method, we introduce a far-field transformation. This transformation is defined independently of the obstacle and augments the near-field operator into the associated far-field operator which does admit a symmetric factorization. This yields a rigorous complete characterization of the obstacle by the factorization method theory and leads to a practical reconstruction algorithm based on the spectral data of the transformed operator. Numerical experiments are presented to demonstrate the effectiveness of the proposed method, with synthetic near-field data generated using the method of fundamental solutions. We also consider reconstructions using only scattered-field measurements generated by point sources, demonstrating the potential for reduced the amount of measured data.
