Stable determination of an impedance obstacle by a single far-field measurement

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems Pub Date : 2024-03-19 DOI:10.1088/1361-6420/ad3087

Huaian Diao, Hongyu Liu, Longyue Tao

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引用次数: 0

Abstract

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in

. The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle’s geometric configurations and the wave field’s vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition.

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通过单次远场测量稳定确定阻抗障碍物

在确定 R2 中的阻抗障碍时，我们建立了对数型的尖锐稳定性估计。障碍物为多边形，表面阻抗参数为非零常数。我们使用单一远场模式建立了稳定性结果，这构成了反向散射理论中的一个长期问题。这是文献中第一个通过单一远场测量确定阻抗障碍物的稳定性结果。同时确定障碍物和边界阻抗的稳定性是根据经典的豪斯多夫距离确定的。为建立上述稳定性结果而开发的数学策略有几项技术创新和发展。首先，稳定性分析是以微观局部的方式围绕角点进行的。其次，我们的稳定性估计在障碍物的几何配置和角点处的波场消失阶数之间建立了明确的关系。第三，我们开发了新颖的误差传播技术，利用阻抗边界条件解决转角处波场的奇异性问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Inverse Problems 数学-物理：数学物理

CiteScore

4.40

自引率

14.30%

发文量

115

审稿时长

2.3 months

期刊介绍： An interdisciplinary journal combining mathematical and experimental papers on inverse problems with theoretical, numerical and practical approaches to their solution. As well as applied mathematicians, physical scientists and engineers, the readership includes those working in geophysics, radar, optics, biology, acoustics, communication theory, signal processing and imaging, among others. The emphasis is on publishing original contributions to methods of solving mathematical, physical and applied problems. To be publishable in this journal, papers must meet the highest standards of scientific quality, contain significant and original new science and should present substantial advancement in the field. Due to the broad scope of the journal, we require that authors provide sufficient introductory material to appeal to the wide readership and that articles which are not explicitly applied include a discussion of possible applications.