A NOVEL INTEGRAL EQUATION FOR SCATTERING BY LOCALLY ROUGH SURFACES AND APPLICATION TO THE INVERSE PROBLEM: THE NEUMANN CASE

doi:10.1137/19M1240745

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	A NOVEL INTEGRAL EQUATION FOR SCATTERING BY LOCALLY ROUGH SURFACES AND APPLICATION TO THE INVERSE PROBLEM: THE NEUMANN CASE
	Qu, Fenglong 1; Zhang, Bo2,3,4 ; Zhang, Haiwen3,5,6
	2019
发表期刊	SIAM JOURNAL ON SCIENTIFIC COMPUTING
ISSN	1064-8275
卷号	41 期号:6 页码:A3673-A3702
摘要	This paper is concerned with direct and inverse scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) on which a Neumann boundary condition is imposed. A novel integral equation formulation is proposed for the direct scattering problem which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners and some closed smooth artificial curve. It is a nontrivial extension of our previous work on direct and inverse scattering by a locally rough surface from the Dirichlet boundary condition to the Neumann boundary condition [SIAM T. Appl. Math., 73 (2013), pp. 1811-1829]. For the Dirichlet boundary condition, the integral equation obtained is uniquely solvable in the space of bounded continuous functions on the bounded curve, and it can be solved efficiently by using the Nystrom method with a graded mesh. However, the Neumann condition case leads to an integral equation which is solvable in the space of squarely integrable functions on the bounded curve rather than in the space of bounded continuous functions, making the integral equation very difficult to solve numerically with the classic and efficient Nystrom method. In this paper, we make use of the recursively compressed inverse preconditioning method developed by Helsing to solve the integral equation which is efficient and capable of dealing with large wave numbers. For the inverse problem, it is proved that the locally rough surface is uniquely determined from a knowledge of the far-field pattern corresponding to incident plane waves. Moreover, based on the novel integral equation formulation, a Newton iteration method is developed to reconstruct the locally rough surface from a knowledge of multiple frequency far-field data. Numerical examples are also provided to illustrate that the reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles.
关键词	integral equation locally rough surface Neumann boundary condition far-field pattern RCIP method Newton iteration method
DOI	10.1137/19M1240745
收录类别	SCI
语种	英语
资助项目	NNSF of China[11871416] ; NNSF of China[11871466] ; NNSF of China[91630309] ; Shandong Provincial Natural Science Foundation[ZR2019MA027] ; Shandong Provincial Natural Science Foundation[ZR2017MA044]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000549131500019
出版者	SIAM PUBLICATIONS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/51817
专题	应用数学研究所
通讯作者	Zhang, Haiwen
作者单位	1.Yantai Univ, Sch Math & Informat Sci, Yantai 264005, Shandong, Peoples R China 2.Chinese Acad Sci, NCMIS, LSEC, Beijing 100190, Peoples R China 3.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China 4.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 5.Chinese Acad Sci, NCMIS, Beijing 100190, Peoples R China 6.Univ Gottingen, Inst Numer & Appl Math, Lotzestr 16-18, D-37083 Gottingen, Germany
推荐引用方式 GB/T 7714	Qu, Fenglong,Zhang, Bo,Zhang, Haiwen. A NOVEL INTEGRAL EQUATION FOR SCATTERING BY LOCALLY ROUGH SURFACES AND APPLICATION TO THE INVERSE PROBLEM: THE NEUMANN CASE[J]. SIAM JOURNAL ON SCIENTIFIC COMPUTING,2019,41(6):A3673-A3702.
APA	Qu, Fenglong,Zhang, Bo,&Zhang, Haiwen.(2019).A NOVEL INTEGRAL EQUATION FOR SCATTERING BY LOCALLY ROUGH SURFACES AND APPLICATION TO THE INVERSE PROBLEM: THE NEUMANN CASE.SIAM JOURNAL ON SCIENTIFIC COMPUTING,41(6),A3673-A3702.
MLA	Qu, Fenglong,et al."A NOVEL INTEGRAL EQUATION FOR SCATTERING BY LOCALLY ROUGH SURFACES AND APPLICATION TO THE INVERSE PROBLEM: THE NEUMANN CASE".SIAM JOURNAL ON SCIENTIFIC COMPUTING 41.6(2019):A3673-A3702.