In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.

翻译：本文针对两个反问题（固定观测方向的多频反源问题和Born反散射问题）提出了一种新的线性采样方法，该方法基于相同数据但采用了不同的数据算子定义。我们证明，当正则化参数趋于零时，相关的正则化线性采样指标收敛于未知函数在某个小邻域内的平均值。通过利用长椭球波函数及其推广形式，我们建立了受其启发、分析并实现的形状识别理论与参数识别理论。进一步，我们提出了基于长椭球函数的线性采样方法实现方案，并通过数值实验展示了该线性采样方法同时重建形状与参数的能力。