Title: A Weighted Sampling Method for Inverse Medium Problem with Limited Aperture Show Chinese title

Title: 一种有限孔径逆介质问题的加权采样方法

Abstract: Inverse medium scattering problems arise in many applications, but in practice, the measurement data are often restricted to a limited aperture by physical or experimental constraints. Classical sampling methods, such as MUSIC and the linear sampling method, are well understood for full-aperture data, yet their performance deteriorates severely under limited-aperture conditions, especially in the presence of noise. We propose a new sampling method tailored to the inverse medium problem with limited-aperture data. The method is motivated by the linear sampling framework and incorporates a weight function into the index function. The weight is designed so that the modified kernel reproduces the full-aperture behavior using only limited data, which both localizes oscillations and improves the conditioning of the far-field system, thereby yielding more accurate and stable reconstructions. We provide a theoretical justification of the method under the Born approximation and an efficient algorithm for computing the weight. Numerical experiments in two and three dimensions demonstrate that the proposed method achieves greater accuracy and robustness than existing sampling-type methods, particularly for noisy, limited-aperture data. Show Chinese abstract

Abstract: 逆介质散射问题出现在许多应用中，但在实际中，由于物理或实验限制，测量数据通常仅限于有限的孔径。 经典的采样方法，如MUSIC和线性采样方法，对于全孔径数据已有很好的理解，但在有限孔径条件下性能会严重下降，尤其是在存在噪声的情况下。 我们提出了一种针对有限孔径数据的逆介质问题的新采样方法。 该方法受到线性采样框架的启发，并将一个权函数引入到指标函数中。 该权函数被设计成仅使用有限数据就能再现全孔径行为，这既局部化了振荡，又改善了远场系统的条件，从而得到了更准确和稳定的重建结果。 我们在Born近似下提供了该方法的理论依据，并给出了计算权函数的高效算法。 二维和三维的数值实验表明，所提出的方法在准确性与鲁棒性方面优于现有的采样型方法，特别是在处理噪声和有限孔径数据时表现更为出色。