用最少散射测量次数稳定确定声硬多面体散射体
Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements
摘要：
本文的目的是建立最佳的稳定性估计确定健全的硬多面体散射体在$\mathbb{R}^N$，$N \geq 2$，通过最少数量的远场测量。这项工作是一个显着的和高度非平凡的扩展的声音软多面体散射体的远场测量的确定的稳定性估计，证明了作者之一，更具挑战性的声音硬的情况下。容许多面体散射体满足Lipschitz型的最小先验假设，并且可以同时包括固体障碍物和屏幕型组件。在这种情况下，我们得到一个稳定性估计与$N$远场测量。这样的估计的重要特征是，我们有一个明确的依赖于参数$h$代表的最小尺寸的细胞形成的可容许的多面体散射体的边界，和连续性的模量，提供的错误是足够小的相对于$h$，不依赖于$h$。如果我们限制到$N=2,3$和多面体障碍物，即多面体，那么我们得到的稳定性估计与较少的测量，即首先与$N-1$测量，然后与一个单一的测量。在这种情况下，对$h$的依赖性不再明确，连续性模量也依赖于$h$。
英文摘要：
The aim of the paper is to establish optimal stability estimates for the determination of sound-hard polyhedral scatterers in $\mathbb{R}^N$, $N \geq 2$, by a minimal number of far-field measurements. This work is a significant and highly nontrivial extension of the stability estimates for the determination of sound-soft polyhedral scatterers by far-field measurements, proved by one of the authors, to the much more challenging sound-hard case. The admissible polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time solid obstacles and screen-type components. In this case we obtain a stability estimate with $N$ far-field measurements. Important features of such an estimate are that we have an explicit dependence on the parameter $h$ representing the minimal size of the cells forming the boundaries of the admissible polyhedral scatterers, and that the modulus of continuity, provided the error is small enough with respect to $h$, does not depend on $h$. If we restrict to $N=2,3$ and to polyhedral obstacles, that is to polyhedra, then we obtain stability estimates with fewer measurements, namely first with $N-1$ measurements and then with a single measurement. In this case the dependence on $h$ is not explicit anymore and the modulus of continuity depends on $h$ as well.
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