THE LINEAR SAMPLING METHOD FOR RECONSTRUCTING A PENETRABLE CAVITY WITH UNKNOWN EXTERNAL OBSTACLES*

doi:10.1007/s10473-023-0216-2

Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (2): 751-776.doi: 10.1007/s10473-023-0216-2

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THE LINEAR SAMPLING METHOD FOR RECONSTRUCTING A PENETRABLE CAVITY WITH UNKNOWN EXTERNAL OBSTACLES^*

Jianguo Ye¹, Guozheng Yan^2,†

1. Research Center of Modern Mathematics and Its Application, School of Mathematics and Statistics, Kashi University, Kashi 844000, China;
2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2020-08-19 Revised:2022-04-26 Online:2023-03-25 Published:2023-04-12
Contact: †Guozheng YAN, E-mail: yangz@ccnu.edu.cn.
About author:Jianguo YE, E-mail: yjgks2001@hotmail.com
Supported by:
The first author was supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region of China (2019D01A05); The second author was supported by the NSFC (11571132).

Abstract

Abstract: We consider the interior inverse scattering problem for recovering the shape of a penetrable partially coated cavity with external obstacles from the knowledge of measured scattered waves due to point sources. In the first part, we obtain the well-posedness of the direct scattering problem by the variational method. In the second part, we establish the mathematical basis of the linear sampling method to recover both the shape of the cavity, and the shape of the external obstacle, however the exterior transmission eigenvalue problem also plays a key role in the discussion of this paper.

Key words: inverse scattering, mixed boundary value problem, exterior transmission eigenvalue, interior measurements, linear sampling method

Jianguo Ye, Guozheng Yan. THE LINEAR SAMPLING METHOD FOR RECONSTRUCTING A PENETRABLE CAVITY WITH UNKNOWN EXTERNAL OBSTACLES^*[J].Acta mathematica scientia,Series B, 2023, 43(2): 751-776.

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References

[1] Hu Y Q, Cakoni F, Liu J J.The inverse scattering problem for a partially coated cavity with interior measurements. Appl Anal, 2014, 93(5): 936-956
[2] Zeng F, Cakoni F, Sun J G.An inverse electromagnetic scattering problem for a cavity. Inverse Probl, 2011, 27(12): 125002
[3] Jakubik P, Potthast R.Testing the integrity of some cavity-the Cauchy problem and the range test. Appl Numer Math, 2008, 58(6): 899-914
[4] Qin H H, Cakoni F.Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem. Inverse Probl, 2011, 27(3): 035005
[5] Qin H H, Colton D.The inverse scattering problem for cavities. Appl Numer Math, 2012, 62(6): 699-708
[6] Qin H H, Colton D.The inverse scattering problem for cavities with impedance boundary condition. Adv Comput Math, 2012, 36(2): 157-174
[7] Qin H H, Liu J C.Reconstruction for cavities with impedance boundary condition. J Integral Equa Appl, 2012, 25(3): 431-454
[8] Qin H H, Liu X X.The interior inverse scattering problem for cavities with an artificial obstacle. Appl Numer Math, 2015, 88(2): 18-30
[9] Liu X D.The factorization method for cavities. Inverse Probl, 2014, 30(1): 015006
[10] Zeng F, Suarez P, Sun J G.A decomposition method for an interior inverse scattering problem. Inverse Probl Imaging, 2013, 7(1): 291-303
[11] Guo J, Ning D, Yan G Z, et al.Reconstruction of a penetrable cavity and the external obstacle. Appl Anal, 2018, 97(10): 1730-1750
[12] Zeng F, Meng S X.The interior inverse electromagnetic scattering for an inhomogeneous cavity. Inverse Probl, 2021, 37(2): 025007
[13] Meng S X, Haddar H, Cakoni F.The factorization method for a cavity in an inhomogeneous medium. Inverse Probl, 2014, 30(4): 045008
[14] Cakoni F, Colton D, Meng S X.The inverse scattering problem for a penetrable cavity with internal measurements//Stefanov P, Vasy A, Zworski M, et al. Inverse Probl Appl. Providence, RI: Amer Math Soc, 2014: 71-88
[15] Liu L H.The inverse scattering problem for a partially coated penetrable cavity with interior measurements. Appl Anal, 2017, 96(5): 844-868
[16] Cakoni F, Colton D.A Qualitative Approach to Inverse Scattering Theory. New York: Springer, 2014
[17] Hörmander L.The Analysis of Linear Partial Differential Operators, III: Pseudo-Differential Operators. Berlin: Springer Science & Business Media, 2007
[18] Colton D. Kress R.Inverse Acoustic and Eletromagnetic Scattering Theory. New York: Springer, 2013
[19] Cakoni F, Colton D, Haddar H.The linear sampling method for anisotropic media. J Comput Appl Math, 2002, 146(2): 285-299
[20] Cakoni F, Haddar H.Transmission eigenvalues in inverse scattering theory. Inside Out II, MSRI Publications, 2012, 60: 527-578
[21] Cakoni F, Colton D, Haddar H.Inverse Scattering Theory and Transmission Eigenvalues. Philadelphia, PA: SIAM, 2016
[22] Isakov V.On uniqueness in th invese transmission scattering problem. Commun Partial Differ Equa, 1990, 15(11): 1565-1587
[23] Isakov V.Inverse Problems for Partial Differential Equations. New York: Springer, 2017
[24] Kirsch A, Grinberg N.The Factorization Method for Inverse Problems. Oxford: Oxford University Press, 2008

THE LINEAR SAMPLING METHOD FOR RECONSTRUCTING A PENETRABLE CAVITY WITH UNKNOWN EXTERNAL OBSTACLES^*

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