Stable Determination of Sound-soft Polyhedral Scatterers by a Single Measurement
Stable Determination of Sound-soft Polyhedral Scatterers by a Single Measurement
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DOI：
10.1512/iumj.2008.57.3217
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发表时间：
2008
影响因子：
1.1
通讯作者：
L. Rondi
中科院分区：
文献类型：
--
作者：
L. Rondi
我们证明了用单次远场测量确定r3中有限数量的声软多面体散射体的最优稳定性估计。可容许的多个多面体散射体满足Lipschitz型的最小先验假设，可以同时包含障碍物、遮挡和更复杂的散射体。我们用一个大小参数h来描述任何多个多面体散射体，这个参数h与它的边界的最小单元尺寸有关。在第一步中，我们证明，如果远场测量的误差e相对于h足够小，那么在多个多面体散射体的豪斯多夫距离上的相应误差可以通过e的显式函数来控制，该函数趋近于零，即e→0 +，这是本质上最优的，尽管是对数的方式。然后，我们展示了如何改进这种稳定性估计，前提是我们将注意力限制在多个多面体障碍物上，并且e相对于h更小。在这种情况下，我们获得了实质上是Holder类型的显式估计。
We prove optimal stability estimates for the determination of a finite number of sound-soft polyhedral scatterers in R 3 by a single far-field measurement. The admissible multiple polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time obstacles, screens and even more complicated scatterers. We characterize any multiple polyhedral scatterer by a size parameter h which is related to the minimal size of the cells of its boundary. In a first step we show that, provided the error e on the far-field measurement is small enough with respect to h, then the corresponding error, in the Hausdorff distance, on the multiple polyhedral scatterer can be controlled by an explicit function of e which approaches zero, as e → 0 + , in an essentially optimal, although logarithmic, way. Then, we show how to improve this stability estimate, provided we restrict our attention to multiple polyhedral obstacles and e is even smaller with respect to h. In this case we obtain an explicit estimate essentially of Holder type.