# American Institute of Mathematical Sciences

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.

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##### 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.
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