In this paper we report our new finding on the linear sampling and factorization methods: in addition to shape identification, the linear sampling and factorization methods have capability in parameter identification. Our demonstration is for shape/parameter identification associated with a restricted Fourier integral operator which arises from the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. Within the framework of linear sampling method, 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. Both the shape and parameter identification theories are general, since the theories allow any general regularization scheme such as the Tikhonov or the singular value cut off regularization. We further propose a prolate-Galerkin formulation of the linear sampling method for implementation and provide numerical experiments to demonstrate how the linear sampling method is capable of reconstructing both the shape and the parameter.

翻译：本文报道了我们在线性采样与分解方法上的新发现：除了形状识别功能外，线性采样与分解方法还具备参数识别能力。我们的论证针对由固定观测方向的多频逆源问题与波恩逆散射问题所导出的限制型傅里叶积分算子，开展形状与参数识别研究。在线性采样法的框架下，我们分别建立了形状识别理论与参数识别理论，这些理论借助长椭球波函数及其推广形式得以激发、分析与实现。两套识别理论均具有普适性，允许采用任意广义正则化方案（如吉洪诺夫正则化或截断奇异值正则化）。我们进一步提出了基于长椭球-伽辽金格式的线性采样法实现方案，并通过数值实验展示了该方法重建形状与参数的能力。