AI 中文总结
研究具有随机各向同性波动的三维光滑星形障碍物逆声散射问题，提出基于蒙特卡罗的多频递归线性化算法，可恢复障碍物几何形状及形状波动场关键统计量，数值实验验证了方法有效性。
AI 中文摘要
在许多实际场景中，散射体形状因各种物理或环境因素呈现不确定的几何变化。对于本质上不适定的逆散射问题，这种几何不确定性对恢复过程可能有不可忽略的影响。本文旨在恢复障碍物几何形状和形状不确定性的统计信息，研究了具有随机各向同性波动的三维光滑星形障碍物的逆声散射问题。提出了一种基于蒙特卡罗的高效多频递归线性化算法，通过对远场算子关于几何参数线性化并采用频率延拓从粗到细尺度恢复未知几何形状。基于重建样本，进一步估计参考几何形状和形状波动场的关键统计量。还证明了远场数据的概率律唯一确定分布中的径向函数，意味着参考形状和相关统计量的唯一性。数值实验证明了该方法在高斯和非高斯随机变化下恢复散射体形状和相关统计信息的有效性。
英文摘要
In many practical scenarios the shapes of scatterers exhibit uncertain geometric variations arising from diverse physical or environmental factors. For inverse scattering problems which are inherently ill-posed, the presence of such geometric uncertainties may have a non-negligible impact on the recovery process. With the aim of recovering both obstacle geometry and statistics of the shape uncertainties, in this paper we study an inverse acoustic scattering problem for three-dimensional smooth star-shaped obstacles with random isotropic fluctuations. We propose an efficient Monte Carlo-based multi-frequency recursive linearization algorithm in which the far-field operator is linearized with respect to the geometry parameters and frequency continuation is employed to recover the unknown geometry from coarse to fine scales. Based on the reconstructed samples, we further estimate the reference geometry and key statistics of the shape fluctuation field including Karhunen--Loève eigenvalues, covariance hyper-parameters for Gaussian perturbations and covariance structure, representative marginal distributions for non-Gaussian perturbations. We also prove that the probability law of the far-field data uniquely determines the radial function in distribution which implies uniqueness of the reference shape and related statistics. Numerical experiments demonstrate the effectiveness of the proposed method in recovering both the scatterer shapes and the associated statistical information under Gaussian and non-Gaussian random variations.
