论文标题
数据驱动的基础，用于重建反向散射中的对比：PICARD标准，规律性，正则化和稳定性
Data-driven basis for reconstructing the contrast in inverse scattering: Picard criterion, regularity, regularization, and stability
论文作者
论文摘要
我们考虑使用出生数据（包括完整的光圈，有限句子和多频数据）重建培养基对比度的反向介质散射。我们根据广义的pr素球波函数和相关本征函数提出了这些反问题的数据驱动基础函数。这种数据驱动的本征函数是傅立叶积分运算符的特征函数。它们在分析上显着扩展到整个空间，是双向的，并且在频带限制功能的类别中是完整的。我们首先建立了使用数据驱动的基础重建对比度的PICARD标准，从数据处理和分析外推的角度来看，重建公式也可以理解。与广义的pr酸球形波函数相关的另一个显着特征是，磁盘的数据驱动基础也是Sturm-Liouville差异操作员的基础。在Sturm-Liouville理论的帮助下，我们估计$ h^s $函数的频谱截止近似值的$ l^2 $近似错误。这会产生嘈杂数据的频谱截止正规化策略，并在完整的孔径情况下（$ 0 <s <1/2 $）中对比度的显式稳定性估计。在有限次数和多频案例中，我们还获得了噪声数据和稳定性估算的光谱截止策略。
We consider the inverse medium scattering of reconstructing the medium contrast using Born data, including the full aperture, limited-aperture, and multi-frequency data. We propose data-driven basis functions for these inverse problems based on the generalized prolate spheroidal wave functions and related eigenfunctions. Such data-driven eigenfunctions are eigenfunctions of a Fourier integral operator; they remarkably extend analytically to the whole space, are doubly orthogonal, and are complete in the class of band-limited functions. We first establish a Picard criterion for reconstructing the contrast using the data-driven basis, where the reconstruction formula can also be understood from the viewpoint of data processing and analytic extrapolation. Another salient feature associated with the generalized prolate spheroidal wave functions is that the data-driven basis for a disk is also a basis for a Sturm-Liouville differential operator. With the help of Sturm-Liouville theory, we estimate the $L^2$ approximation error for a spectral cutoff approximation of $H^s$ functions. This yields a spectral cutoff regularization strategy for noisy data and an explicit stability estimate for contrast in $H^s$ ($0<s<1/2$) in the full aperture case. In the limited-aperture and multi-frequency cases, we also obtain spectral cutoff regularization strategies for noisy data and stability estimates for a class of contrast.
