标题： 多频段有限孔径远场数据的反源问题增加稳定性 显示英文标题

标题： Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

摘要： 我们研究在多个波数下的有限孔径远场数据下，亥姆霍兹方程逆源问题的增加稳定性。测量数据由远场模式$u^\infity(\hat{x},k)$给出，对于固定方向$\hat{x}$的某个邻域内的所有观测方向以及属于有限区间$(0,K)$的所有波数 k。在本文中，我们讨论了相对于波数区间$K>1$宽度的增加稳定性。在三维情况下，我们从远场数据建立了源函数的$L^2$-范数和$H^{-1}$-范数的稳定性估计。逆源问题的不适定性在增加波数带 K 时表现为 Hölder 类型。我们还讨论了在固定方向下远场数据关于波数的解析延拓方法。 显示英文摘要

摘要： We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns $u^\infity(\hat{x},k)$ for all observation directions in some neighborhood of a fixed direction $\hat{x}$ and for all wave numbers k belonging to a finite interval $(0,K)$. In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval $K>1$. In three dimensions we establish stability estimates of the $L^2$-norm and $H^{-1}$-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of H\"older type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.