# American Institute of Mathematical Sciences

August  2016, 10(3): 855-868. doi: 10.3934/ipi.2016024

## The reciprocity gap method for a cavity in an inhomogeneous medium

 1 Postdoctoral Station of Optical Engineering, Institute of Computing and Data Sciences, College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China 2 Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190 3 Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931 4 Institute of Computing and Data Sciences, College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received  April 2015 Revised  February 2016 Published  August 2016

We consider an interior inverse medium problem of reconstructing the shape of a cavity. Both the measurement locations and point sources are inside the cavity. Due to the lack of a priori knowledge of physical prosperities of the medium inside the cavity and to avoid the computation of background Green's functions, the reciprocity gap method is employed. We prove the related theory and present some numerical examples for validation.
Citation: Fang Zeng, Xiaodong Liu, Jiguang Sun, Liwei Xu. The reciprocity gap method for a cavity in an inhomogeneous medium. Inverse Problems and Imaging, 2016, 10 (3) : 855-868. doi: 10.3934/ipi.2016024
##### References:
 [1] T. Arens, Why linear sampling works, Inverse Problems, 20 (2004), 163-173. doi: 10.1088/0266-5611/20/1/010. [2] F. Cakoni and D. Colton, Qualitative Approach to Inverse Scattering Theory, Springer, Berlin, 2014. doi: 10.1007/978-1-4612-0873-0. [3] F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, AMS Contemporary Mathematics, 615 (2014), 71-88. doi: 10.1090/conm/615/12246. [4] F. Cakoni, D. Colton and H. Haddar, The linear sampling method for anisotropic media, J. Comput. Appl. Math., 146 (2002), 285-299. doi: 10.1016/S0377-0427(02)00361-8. [5] D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inverse Problems, 21 (2005), 383-398. doi: 10.1088/0266-5611/21/1/023. [6] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1983. doi: 10.1007/978-1-4612-0873-0. [7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $2^{nd}$ Edition, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-0873-0. [8] D. Colton and B. D. Sleeman, An approximation property of importance in inverse scattering theory, Proc. Edinburgh Math. Soc., 44 (2001), 449-454. doi: 10.1017/S0013091500000626. [9] M. D. Cristo and J. Sun, An inverse scattering problem for a partially coated buried obstacle, Inverse Problems, 22 (2006), 2331-2350. doi: 10.1088/0266-5611/22/6/025. [10] G. Hu and X. Liu, Unique determination of balls and polyhedral scatters with a single point source wave, Inverse Problems, 30 (2014), 065010, 14pp. doi: 10.1088/0266-5611/30/6/065010. [11] Y. Hu, F. Cakoni and J. Liu, The inverse scattering problem for a partially coated cavity with interior measurements, Applicable Analysis, 93 (2013), 936-956. doi: 10.1080/00036811.2013.801458. [12] P. Jakubik and R. Potthast, Testing the integrity of some cavity - the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914. doi: 10.1016/j.apnum.2007.04.007. [13] J. Li, H. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009), 4013-4040. doi: 10.1137/080734170. [14] P. Li and Y. Wang, Near-field imaging of interior cavities, Commun. Comput. Phys., 17 (2015), 542-563. doi: 10.4208/cicp.010414.250914a. [15] X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18pp. doi: 10.1088/0266-5611/30/1/015006. [16] P. Monk and J. Sun, Inverse scattering using finite elements and gap reciprocity, Inverse Problems and Imaging, 1 (2007), 643-660. doi: 10.3934/ipi.2007.1.643. [17] P. Monk and V. Selgas, Sampling type methods for an inverse waveguide problem, Inverse Problems and Imaging, 6 (2012), 709-747. doi: 10.3934/ipi.2012.6.709. [18] S. 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. [19] M. Powell, Approximation Theory and Methods, Cambridge University Press, Cambridge, 1981. doi: 10.1007/978-1-4612-0873-0. [20] H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17pp. doi: 10.1088/0266-5611/27/3/035005. [21] H. Qin and D. Colton, The inverse scattering problem for cavities, Applied Numerical Mathematics, 62 (2012), 699-708. doi: 10.1016/j.apnum.2010.10.011. [22] H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Advances in Computational Mathematics, 36 (2012), 157-174. doi: 10.1007/s10444-011-9179-2. [23] H. Qin and X. Liu, The interior inverse scattering problem for cavities with an artificial obstacle, Applied Numerical Mathematics, 88 (2015), 18-30. doi: 10.1016/j.apnum.2014.10.002. [24] G. Uhlmann, Inverse scattering in anisotropic media, in Chapter Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 235-251. doi: 10.1007/978-1-4612-0873-0. [25] F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002, 17pp. doi: 10.1088/0266-5611/27/12/125002. [26] F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Problems and Imaging, 7 (2013), 291-303. doi: 10.3934/ipi.2013.7.291. [27] F. Zeng, X. Liu, J. Sun and L. Xu, Reciprocity gap method for an interior inverse scattering problem, Journal of Inverse and Ill-posed Problems, online, (2016). doi: 10.1515/jiip-2015-0064.

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##### References:
 [1] T. Arens, Why linear sampling works, Inverse Problems, 20 (2004), 163-173. doi: 10.1088/0266-5611/20/1/010. [2] F. Cakoni and D. Colton, Qualitative Approach to Inverse Scattering Theory, Springer, Berlin, 2014. doi: 10.1007/978-1-4612-0873-0. [3] F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, AMS Contemporary Mathematics, 615 (2014), 71-88. doi: 10.1090/conm/615/12246. [4] F. Cakoni, D. Colton and H. Haddar, The linear sampling method for anisotropic media, J. Comput. Appl. Math., 146 (2002), 285-299. doi: 10.1016/S0377-0427(02)00361-8. [5] D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inverse Problems, 21 (2005), 383-398. doi: 10.1088/0266-5611/21/1/023. [6] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1983. doi: 10.1007/978-1-4612-0873-0. [7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $2^{nd}$ Edition, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-0873-0. [8] D. Colton and B. D. Sleeman, An approximation property of importance in inverse scattering theory, Proc. Edinburgh Math. Soc., 44 (2001), 449-454. doi: 10.1017/S0013091500000626. [9] M. D. Cristo and J. Sun, An inverse scattering problem for a partially coated buried obstacle, Inverse Problems, 22 (2006), 2331-2350. doi: 10.1088/0266-5611/22/6/025. [10] G. Hu and X. Liu, Unique determination of balls and polyhedral scatters with a single point source wave, Inverse Problems, 30 (2014), 065010, 14pp. doi: 10.1088/0266-5611/30/6/065010. [11] Y. Hu, F. Cakoni and J. Liu, The inverse scattering problem for a partially coated cavity with interior measurements, Applicable Analysis, 93 (2013), 936-956. doi: 10.1080/00036811.2013.801458. [12] P. Jakubik and R. Potthast, Testing the integrity of some cavity - the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914. doi: 10.1016/j.apnum.2007.04.007. [13] J. Li, H. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009), 4013-4040. doi: 10.1137/080734170. [14] P. Li and Y. Wang, Near-field imaging of interior cavities, Commun. Comput. Phys., 17 (2015), 542-563. doi: 10.4208/cicp.010414.250914a. [15] X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18pp. doi: 10.1088/0266-5611/30/1/015006. [16] P. Monk and J. Sun, Inverse scattering using finite elements and gap reciprocity, Inverse Problems and Imaging, 1 (2007), 643-660. doi: 10.3934/ipi.2007.1.643. [17] P. Monk and V. Selgas, Sampling type methods for an inverse waveguide problem, Inverse Problems and Imaging, 6 (2012), 709-747. doi: 10.3934/ipi.2012.6.709. [18] S. 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. [19] M. Powell, Approximation Theory and Methods, Cambridge University Press, Cambridge, 1981. doi: 10.1007/978-1-4612-0873-0. [20] H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17pp. doi: 10.1088/0266-5611/27/3/035005. [21] H. Qin and D. Colton, The inverse scattering problem for cavities, Applied Numerical Mathematics, 62 (2012), 699-708. doi: 10.1016/j.apnum.2010.10.011. [22] H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Advances in Computational Mathematics, 36 (2012), 157-174. doi: 10.1007/s10444-011-9179-2. [23] H. Qin and X. Liu, The interior inverse scattering problem for cavities with an artificial obstacle, Applied Numerical Mathematics, 88 (2015), 18-30. doi: 10.1016/j.apnum.2014.10.002. [24] G. Uhlmann, Inverse scattering in anisotropic media, in Chapter Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 235-251. doi: 10.1007/978-1-4612-0873-0. [25] F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002, 17pp. doi: 10.1088/0266-5611/27/12/125002. [26] F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Problems and Imaging, 7 (2013), 291-303. doi: 10.3934/ipi.2013.7.291. [27] F. Zeng, X. Liu, J. Sun and L. Xu, Reciprocity gap method for an interior inverse scattering problem, Journal of Inverse and Ill-posed Problems, online, (2016). doi: 10.1515/jiip-2015-0064.
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