February  2016, 10(1): 263-279. doi: 10.3934/ipi.2016.10.263

The factorization method for a partially coated cavity in inverse scattering

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan, China, China

Received  January 2015 Revised  July 2015 Published  February 2016

We consider the interior inverse scattering problem of recovering the shape of an impenetrable partially coated cavity. The scattered fields incited by point source waves are measured on a closed curve inside the cavity. We prove the validity of the factorization method for reconstructing the shape of the cavity. However, we are not able to apply the basic theorem introduced by Kirsch and Grinberg to treat the key operator directly, and some auxiliary operators have to be considered. In this paper, we provide theoretical validation of the factorization method to the problem, and some numerical results are presented to show the viability of our method.
Citation: Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems and Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263
References:
[1]

T. Angell and R. Kleinmann, The Helmholtz equation with $L^{2}$-boundary values, SIAM J. Math. Anal., 16 (1985), 259-278. doi: 10.1137/0516020.

[2]

T. Angell and A. Kirsch, Optimization Methods in Electromagnetic Radiation, Springer-Verlag, New York, 2004. doi: 10.1007/b97629.

[3]

O. Bondarenko and X. Liu, The factorization method for inverse obstacle scattering with conductive boundary condition, Inverse Problems, 29 (2013), 095021, 25pp. doi: 10.1088/0266-5611/29/9/095021.

[4]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Problems and Imaging, 7 (2013), 1123-1138. doi: 10.3934/ipi.2013.7.1123.

[5]

F. Cakoni, D. Colton and P. Monk, The direct and inverse scattering problems for partially coated obstacles, Inverse Problems, 17 (2001), 1997-2015. doi: 10.1088/0266-5611/17/6/327.

[6]

F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295. doi: 10.1088/0266-5611/19/2/303.

[7]

F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, AMS Contemp. Math., 615 (2014), 71-88. doi: 10.1090/conm/615/12246.

[8]

M. Chamaillard, N. Chaulet and H. Haddar, Analysis of the factorization method for a general class of boundary conditions, J. of Inverse and Ill-posed Problems, 22 (2014), 643-670. doi: 10.1515/jip-2013-0013.

[9]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.

[10]

K. Daisuke, Error estimates of the DtN finite element method for the exterior Helmholtz Problem, J. Comp. Appl. Math., 200 (2007), 21-31. doi: 10.1016/j.cam.2005.12.004.

[11]

N. I. Grinberg and A. Kirsch, The factorization method for obstacles with a priori separated sound-soft and sound-hard parts, Math. Comput. Simulation, 66 (2004), 267-279. doi: 10.1016/j.matcom.2004.02.011.

[12]

N. I. Grinberg, The operator factorazition method in inverse obstacle scattering, Integral Equations and Operator Theory, 54 (2006), 333-348. doi: 10.1007/s00020-004-1355-z.

[13]

Y. Hu, F. Cakoni and J. Liu, The inverse problem for a partially coated cavity with interior measurements, Appl. Anal., 93 (2014), 936-956. doi: 10.1080/00036811.2013.801458.

[14]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.

[15]

A. Kirsch, Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, 15 (1999), 413-429. doi: 10.1088/0266-5611/15/2/005.

[16]

A. Kirsch and N. I. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, New York, 2008.

[17]

A. Kirsch and X. Liu, Direct and inverse acoustic scattering by a mixed-type scatterer, Inverse Problems, 29 (2013), 065005, 19pp. doi: 10.1088/0266-5611/29/6/065005.

[18]

A. Kirsch and X. Liu, A modification of the factorization method for the classical acoustic inverse scattering problem, Inverse Problems, 30 (2014), 035013, 14pp. doi: 10.1088/0266-5611/30/3/035013.

[19]

R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comput. Appl. Math., 61 (1995), 345-360. doi: 10.1016/0377-0427(94)00073-7.

[20]

R. Kress and K. M. Lee, Integral equation method for scattering from an impedance crack, J. Comp. Appl. Math., 161 (2003), 161-177. doi: 10.1016/S0377-0427(03)00586-7.

[21]

X. D. Liu, The factorization method for cavities, Inverse problems, 30 (2014), 015006, 18pp. doi: 10.1088/0266-5611/30/1/015006.

[22]

W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

[23]

S. X. Meng, H. Haddar and F. Cakoni, The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30 (2014), 045008, 20pp. doi: 10.1088/0266-5611/30/4/045008.

[24]

L. Mönch, On the inverse acoustic scattering problem by an open arc: The sound-hard case, Inverse Problems, 13 (1997), 1379-1392. doi: 10.1088/0266-5611/13/5/017.

[25]

H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17pp. doi: 10.1088/0266-5611/27/3/035005.

[26]

H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699-708. doi: 10.1016/j.apnum.2010.10.011.

[27]

J. Yang, B. Zhang and H. Zhang, The factorization method for reconstructing a penetrble obstacle with unknown buried objects, SIAM. J. Appl. Math., 73 (2013), 617-635. doi: 10.1137/120883724.

[28]

J. Yang, B. Zhang and H. Zhang, Reconstruction of complex obstacles with generalized impedance boundary conditions from far-field data, SIAM J. Appl. Math., 74 (2014), 106-124. doi: 10.1137/130921350.

[29]

F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002. doi: 10.1088/0266-5611/27/12/125002.

[30]

F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior scattering problem, Inverse Problems and Imaging, 7 (2013), 291-303. doi: 10.3934/ipi.2013.7.291.

show all references

References:
[1]

T. Angell and R. Kleinmann, The Helmholtz equation with $L^{2}$-boundary values, SIAM J. Math. Anal., 16 (1985), 259-278. doi: 10.1137/0516020.

[2]

T. Angell and A. Kirsch, Optimization Methods in Electromagnetic Radiation, Springer-Verlag, New York, 2004. doi: 10.1007/b97629.

[3]

O. Bondarenko and X. Liu, The factorization method for inverse obstacle scattering with conductive boundary condition, Inverse Problems, 29 (2013), 095021, 25pp. doi: 10.1088/0266-5611/29/9/095021.

[4]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Problems and Imaging, 7 (2013), 1123-1138. doi: 10.3934/ipi.2013.7.1123.

[5]

F. Cakoni, D. Colton and P. Monk, The direct and inverse scattering problems for partially coated obstacles, Inverse Problems, 17 (2001), 1997-2015. doi: 10.1088/0266-5611/17/6/327.

[6]

F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295. doi: 10.1088/0266-5611/19/2/303.

[7]

F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, AMS Contemp. Math., 615 (2014), 71-88. doi: 10.1090/conm/615/12246.

[8]

M. Chamaillard, N. Chaulet and H. Haddar, Analysis of the factorization method for a general class of boundary conditions, J. of Inverse and Ill-posed Problems, 22 (2014), 643-670. doi: 10.1515/jip-2013-0013.

[9]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.

[10]

K. Daisuke, Error estimates of the DtN finite element method for the exterior Helmholtz Problem, J. Comp. Appl. Math., 200 (2007), 21-31. doi: 10.1016/j.cam.2005.12.004.

[11]

N. I. Grinberg and A. Kirsch, The factorization method for obstacles with a priori separated sound-soft and sound-hard parts, Math. Comput. Simulation, 66 (2004), 267-279. doi: 10.1016/j.matcom.2004.02.011.

[12]

N. I. Grinberg, The operator factorazition method in inverse obstacle scattering, Integral Equations and Operator Theory, 54 (2006), 333-348. doi: 10.1007/s00020-004-1355-z.

[13]

Y. Hu, F. Cakoni and J. Liu, The inverse problem for a partially coated cavity with interior measurements, Appl. Anal., 93 (2014), 936-956. doi: 10.1080/00036811.2013.801458.

[14]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.

[15]

A. Kirsch, Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, 15 (1999), 413-429. doi: 10.1088/0266-5611/15/2/005.

[16]

A. Kirsch and N. I. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, New York, 2008.

[17]

A. Kirsch and X. Liu, Direct and inverse acoustic scattering by a mixed-type scatterer, Inverse Problems, 29 (2013), 065005, 19pp. doi: 10.1088/0266-5611/29/6/065005.

[18]

A. Kirsch and X. Liu, A modification of the factorization method for the classical acoustic inverse scattering problem, Inverse Problems, 30 (2014), 035013, 14pp. doi: 10.1088/0266-5611/30/3/035013.

[19]

R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comput. Appl. Math., 61 (1995), 345-360. doi: 10.1016/0377-0427(94)00073-7.

[20]

R. Kress and K. M. Lee, Integral equation method for scattering from an impedance crack, J. Comp. Appl. Math., 161 (2003), 161-177. doi: 10.1016/S0377-0427(03)00586-7.

[21]

X. D. Liu, The factorization method for cavities, Inverse problems, 30 (2014), 015006, 18pp. doi: 10.1088/0266-5611/30/1/015006.

[22]

W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

[23]

S. X. Meng, H. Haddar and F. Cakoni, The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30 (2014), 045008, 20pp. doi: 10.1088/0266-5611/30/4/045008.

[24]

L. Mönch, On the inverse acoustic scattering problem by an open arc: The sound-hard case, Inverse Problems, 13 (1997), 1379-1392. doi: 10.1088/0266-5611/13/5/017.

[25]

H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17pp. doi: 10.1088/0266-5611/27/3/035005.

[26]

H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699-708. doi: 10.1016/j.apnum.2010.10.011.

[27]

J. Yang, B. Zhang and H. Zhang, The factorization method for reconstructing a penetrble obstacle with unknown buried objects, SIAM. J. Appl. Math., 73 (2013), 617-635. doi: 10.1137/120883724.

[28]

J. Yang, B. Zhang and H. Zhang, Reconstruction of complex obstacles with generalized impedance boundary conditions from far-field data, SIAM J. Appl. Math., 74 (2014), 106-124. doi: 10.1137/130921350.

[29]

F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002. doi: 10.1088/0266-5611/27/12/125002.

[30]

F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior scattering problem, Inverse Problems and Imaging, 7 (2013), 291-303. doi: 10.3934/ipi.2013.7.291.

[1]

Fenglong Qu, Jiaqing Yang. On recovery of an inhomogeneous cavity in inverse acoustic scattering. Inverse Problems and Imaging, 2018, 12 (2) : 281-291. doi: 10.3934/ipi.2018012

[2]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems and Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

[3]

Yunwen Yin, Weishi Yin, Pinchao Meng, Hongyu Liu. The interior inverse scattering problem for a two-layered cavity using the Bayesian method. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021069

[4]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[5]

Guanqiu Ma, Guanghui Hu. Factorization method for inverse time-harmonic elastic scattering with a single plane wave. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022050

[6]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems and Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

[7]

Xiaoxu Xu, Bo Zhang, Haiwen Zhang. Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Problems and Imaging, 2020, 14 (3) : 489-510. doi: 10.3934/ipi.2020023

[8]

Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems and Imaging, 2021, 15 (5) : 1247-1267. doi: 10.3934/ipi.2021036

[9]

Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems and Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951

[10]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems and Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211

[11]

Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems and Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103

[12]

Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems and Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016

[13]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[14]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[15]

Lu Zhao, Heping Dong, Fuming Ma. Inverse obstacle scattering for acoustic waves in the time domain. Inverse Problems and Imaging, 2021, 15 (5) : 1269-1286. doi: 10.3934/ipi.2021037

[16]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems and Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004

[17]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351

[18]

Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062

[19]

Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029

[20]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems and Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (367)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]