Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Recovery of multiple obstacles by probe method

Authors: Jin Cheng, Jijun Liu, Gen Nakamura and Shengzhang Wang
Journal: Quart. Appl. Math. 67 (2009), 221-247
MSC (2000): Primary 35R30, 35J05, 76Q05
DOI: https://doi.org/10.1090/S0033-569X-09-01101-0
Published electronically: March 27, 2009
MathSciNet review: 2514633
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Abstract: We consider an inverse scattering problem for multiple obstacles $ D=\bigcup_{j=1}^ND_j\subset {R}^3$ with different types of boundary for $ D_j$. By constructing an indicator function from the far-field pattern of the scattered wave, we can firstly reconstruct the shape of all obstacles, then identify the type of boundary for each obstacle, as well as the boundary impedance in the case that obstacles have the Robin-type boundary condition. The novelty of our probe method compared with the existing probe method is that we succeeded in identifying the type of boundary condition for multiple obstacles by analyzing the behavior of both the imaginary part and the real part of the indicator function. The numerical realizations are given to show the performance of this inversion method.

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Additional Information

Jin Cheng
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
Email: jcheng@fudan.edu.cn

Jijun Liu
Affiliation: Department of Mathematics, Southeast University, Nanjing 210096, People’s Republic of China
Email: jjliu@seu.edu.cn

Gen Nakamura
Affiliation: Department of Mathematics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan
Email: gnaka@math.sci.hokudai.ac.jp

Shengzhang Wang
Affiliation: Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433, People’s Republic of China
Email: szwang@fudan.edu.cn

DOI: https://doi.org/10.1090/S0033-569X-09-01101-0
Keywords: Inverse scattering, probe method, uniqueness, indicator, numerics
Received by editor(s): September 7, 2007
Published electronically: March 27, 2009
Additional Notes: The first author is partly supported by NSFC (No.9207011148, No.3007011043).
The second author is supported by NSFC (No.10371018).
The third author is partly supported by Grant-in-Aid for scientific research, Ministry of Education, Science and Culture, Japan (No.12640153).
The fourth author is the corresponding author for this paper.
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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