Inverse elastic scattering problems with phaseless far field data

doi:10.1088/1361-6420/ab2a35

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	Inverse elastic scattering problems with phaseless far field data
	Ji,Xia1 ; Liu,Xiaodong2
	2019-10-03
发表期刊	Inverse Problems
ISSN	0266-5611
卷号	35 期号:11
摘要	Abstract This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for inverse elastic scattering problems with phaseless far field data. The phaseless far field data is closely related to the outward energy flux, which is easily measured in practice. Systematically, we study two basic models, i.e. inverse scattering of plane waves by rigid bodies and inverse scattering of sources with compact support. For both models, we show that the phaseless far field data is invariant under translation of the underlying scattering objects, which implies that the location of the objects can not be uniquely recovered by the data. To solve this problem, we consider simultaneously the incident point sources with one fixed source point and at most three scattering strengths. With this technique, we establish some uniqueness results for source scattering problem with multi-frequency phaseless far field data. Furthermore, a fast and stable phase retrieval approach is proposed based on a simple geometric result which provides a stable reconstruction of a point in the plane from three distances to given points. Difficulties arise for inverse scattering by rigid bodies due to the additional unknown far field pattern of the point sources. To overcome this difficulty, we introduce an artificial rigid body into the system and show that the underlying rigid bodies can be uniquely determined by the corresponding phaseless far field data at a fixed frequency. Noting that the far field pattern of the scattered field corresponding to point sources is very small if the source point is far away from the scatterers, we propose an appropriate phase retrieval method for obstacle scattering problems, without using the artificial rigid body. Finally, we propose several sampling methods for shape reconstruction with phaseless far field data directly. For inverse obstacle scattering problems, two different direct sampling methods are proposed with data at a fixed frequency. For inverse source scattering problems, we introduce two direct sampling methods for source supports with sparse multi-frequency data. The phase retrieval techniques are also combined with the classical sampling methods for the shape reconstructions. Extended numerical examples in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed phase retrieval techniques and sampling methods.
关键词	elastic scattering phaseless far field data uniqueness phase retrieval direct sampling methods
DOI	10.1088/1361-6420/ab2a35
语种	英语
WOS记录号	IOP:0266-5611-35-11-ab2a35
出版者	IOP Publishing
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/33327
专题	应用数学研究所计算数学与科学工程计算研究所
作者单位	1.LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 2.Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, People’s Republic of China
推荐引用方式 GB/T 7714	Ji,Xia,Liu,Xiaodong. Inverse elastic scattering problems with phaseless far field data[J]. Inverse Problems,2019,35(11).
APA	Ji,Xia,&Liu,Xiaodong.(2019).Inverse elastic scattering problems with phaseless far field data.Inverse Problems,35(11).
MLA	Ji,Xia,et al."Inverse elastic scattering problems with phaseless far field data".Inverse Problems 35.11(2019).