# American Institute of Mathematical Sciences

May  2013, 7(2): 545-563. doi: 10.3934/ipi.2013.7.545

## Imaging acoustic obstacles by singular and hypersingular point sources

 1 Faculty of Science, South University of Science and Technology of China, Shenzhen, 518055, China 2 Department of Mathematics and Statistics, University of North Carolina, Charlotte, NC 28223, United States 3 Institute of Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China 4 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Received  September 2012 Revised  October 2012 Published  May 2013

We investigate a qualitative method for imaging acoustic obstacles in two and three dimensions by boundary measurements corresponding to hypersingular point sources. Rigorous mathematical justification of the imaging method is established, and numerical experiments are presented to illustrate the effectiveness of the proposed imaging scheme.
Citation: Jingzhi Li, Hongyu Liu, Hongpeng Sun, Jun Zou. Imaging acoustic obstacles by singular and hypersingular point sources. Inverse Problems and Imaging, 2013, 7 (2) : 545-563. doi: 10.3934/ipi.2013.7.545
##### References:
 [1] H. Ammari, "An Introduction to Mathematics of Emerging Biomedical Imaging," Mathematics and Applications, 62, Springer-Verlag, Berlin, 2008. [2] H. Ammari, J. Garnier, H. Kang, M. Lim and K. Solna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564-600. doi: 10.1137/10080631X. [3] H. Ammari, R. Griesmaier and M. Hanke, Identification of small inhomogeneities: Asymptotic factorization, Math. Comp., 76 (2007), 1425-1448. doi: 10.1090/S0025-5718-07-01946-1. [4] H. Ammari and H. Kang, "Expansion Methods," Handbook of Mathematical Methods in Imaging, Springer-Verlag, New York, 2011, 447-499. doi: 10.1090/conm/548. [5] H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements," Springer-Verlag, Berlin Heidelberg, 2004. doi: 10.1007/b98245. [6] H. Ammari, H. Kang, H. Lee and W. K. Park, Asymptotic imaging of perfectly conducting cracks, SIAM J. Sci. Comput., 32 (2010), 894-922. doi: 10.1137/090749013. [7] T. Arens, Why the linear sampling method works, Inverse Problems, 20 (2004), 163-173. doi: 10.1088/0266-5611/20/1/010. [8] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory," Springer-Verlag, Berlin Heidelberg, 2006. [9] M. Cheney, The linear sampling method and the MUSIC algorithm, Inverse Problems, 17 (2001), 591-595. doi: 10.1088/0266-5611/17/4/301. [10] W. C. Chew, "Waves and Fields In Inhomogenenous Media," Van Nostrand Reinhold, 1990. [11] D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic inverse scattering theory, Inverse Problems, 19 (2003), S105-s137. doi: 10.1088/0266-5611/19/6/057. [12] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [13] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," $2^{nd}$ edition, Springer-Verlag, New York, 1998. [14] D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-r66. doi: 10.1088/0266-5611/22/3/R01. [15] D. Colton and P. Monk, The numerical solution of the three-dimensional inverse scattering problem for time harmonic acoustic waves, SIAM J. Sci. Stat. Comput., 8 (1987), 278-291. doi: 10.1137/0908035. [16] D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves, SIAM J. Appl. Math., 58 (1998), 926-941. doi: 10.1137/S0036139996308005. [17] B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050. doi: 10.1088/0266-5611/21/6/015. [18] T. Ide, H. Isozaki, S. Nakata, S. Siltanen and G. Uhlmann, Probing for electrical inclusions with complex spherical waves, Commu. Pure. Appl. Math., 60 (2007), 1415-1442. doi: 10.1002/cpa.20194. [19] M. Ikehata and H. Itou, Extracting the support function of a cavity in an isotropic elastic body from a single set of boundary data, Inverse Problems, 25 (2009), 105005. doi: 10.1088/0266-5611/25/10/105005. [20] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241. doi: 10.1088/0266-5611/15/5/308. [21] M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inv. Ill-posed Problems, 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255. [22] V. Isakov, "Inverse Problems for Partial Differential Equations," $2^{nd}$ edition, Springer-Verlag, New York, 2006. [23] A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, 2008. [24] J. Z. Li, H. Y. Liu and J. Zou, Multilevel linear sampling method for inverse scattering problems, SIAM J. Sci. Comp., 30 (2008), 1228-1250. doi: 10.1137/060674247. [25] J. Z. Li, H. Y. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comp., 31 (2009), 4013-4040. doi: 10.1137/080734170. [26] J. Z. Li, H. Y. Liu, H. P. Sun and J. Zou, Reconstructing acoustic obstacle by planar and cylindrical waves, J. Math. Phys., 53 (2012), 103705 . doi: 10.1063/1.4751282. [27] W. Mclean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, 2000. [28] G. Nakamura and K. Yoshida, Identification of a non-convex obstacle for acoustical scattering, J. Inv. Ill-posed Problems, 15 (2007), 611-624. doi: 10.1515/jiip.2007.034. [29] R. Potthast, A fast new method to solve inverse scattering problem, Inverse Problem, 12 (1996), 731-742. doi: 10.1088/0266-5611/12/5/014. [30] R. Potthast, "Point Sources and Multipoles in Inverse Scattering Theorey," Chapman& Hall/CRC Research Notes in Math., 2001. doi: 10.1201/9781420035483. [31] R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), R1-R47. doi: 10.1088/0266-5611/22/2/R01. [32] P. Martin, "Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles," Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2006 doi: 10.1017/CBO9780511735110.

show all references

##### References:
 [1] H. Ammari, "An Introduction to Mathematics of Emerging Biomedical Imaging," Mathematics and Applications, 62, Springer-Verlag, Berlin, 2008. [2] H. Ammari, J. Garnier, H. Kang, M. Lim and K. Solna, Multistatic imaging of extended targets, SIAM J. Imaging Sci., 5 (2012), 564-600. doi: 10.1137/10080631X. [3] H. Ammari, R. Griesmaier and M. Hanke, Identification of small inhomogeneities: Asymptotic factorization, Math. Comp., 76 (2007), 1425-1448. doi: 10.1090/S0025-5718-07-01946-1. [4] H. Ammari and H. Kang, "Expansion Methods," Handbook of Mathematical Methods in Imaging, Springer-Verlag, New York, 2011, 447-499. doi: 10.1090/conm/548. [5] H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements," Springer-Verlag, Berlin Heidelberg, 2004. doi: 10.1007/b98245. [6] H. Ammari, H. Kang, H. Lee and W. K. Park, Asymptotic imaging of perfectly conducting cracks, SIAM J. Sci. Comput., 32 (2010), 894-922. doi: 10.1137/090749013. [7] T. Arens, Why the linear sampling method works, Inverse Problems, 20 (2004), 163-173. doi: 10.1088/0266-5611/20/1/010. [8] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory," Springer-Verlag, Berlin Heidelberg, 2006. [9] M. Cheney, The linear sampling method and the MUSIC algorithm, Inverse Problems, 17 (2001), 591-595. doi: 10.1088/0266-5611/17/4/301. [10] W. C. Chew, "Waves and Fields In Inhomogenenous Media," Van Nostrand Reinhold, 1990. [11] D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic inverse scattering theory, Inverse Problems, 19 (2003), S105-s137. doi: 10.1088/0266-5611/19/6/057. [12] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [13] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," $2^{nd}$ edition, Springer-Verlag, New York, 1998. [14] D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-r66. doi: 10.1088/0266-5611/22/3/R01. [15] D. Colton and P. Monk, The numerical solution of the three-dimensional inverse scattering problem for time harmonic acoustic waves, SIAM J. Sci. Stat. Comput., 8 (1987), 278-291. doi: 10.1137/0908035. [16] D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves, SIAM J. Appl. Math., 58 (1998), 926-941. doi: 10.1137/S0036139996308005. [17] B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050. doi: 10.1088/0266-5611/21/6/015. [18] T. Ide, H. Isozaki, S. Nakata, S. Siltanen and G. Uhlmann, Probing for electrical inclusions with complex spherical waves, Commu. Pure. Appl. Math., 60 (2007), 1415-1442. doi: 10.1002/cpa.20194. [19] M. Ikehata and H. Itou, Extracting the support function of a cavity in an isotropic elastic body from a single set of boundary data, Inverse Problems, 25 (2009), 105005. doi: 10.1088/0266-5611/25/10/105005. [20] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241. doi: 10.1088/0266-5611/15/5/308. [21] M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inv. Ill-posed Problems, 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255. [22] V. Isakov, "Inverse Problems for Partial Differential Equations," $2^{nd}$ edition, Springer-Verlag, New York, 2006. [23] A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, 2008. [24] J. Z. Li, H. Y. Liu and J. Zou, Multilevel linear sampling method for inverse scattering problems, SIAM J. Sci. Comp., 30 (2008), 1228-1250. doi: 10.1137/060674247. [25] J. Z. Li, H. Y. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comp., 31 (2009), 4013-4040. doi: 10.1137/080734170. [26] J. Z. Li, H. Y. Liu, H. P. Sun and J. Zou, Reconstructing acoustic obstacle by planar and cylindrical waves, J. Math. Phys., 53 (2012), 103705 . doi: 10.1063/1.4751282. [27] W. Mclean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, 2000. [28] G. Nakamura and K. Yoshida, Identification of a non-convex obstacle for acoustical scattering, J. Inv. Ill-posed Problems, 15 (2007), 611-624. doi: 10.1515/jiip.2007.034. [29] R. Potthast, A fast new method to solve inverse scattering problem, Inverse Problem, 12 (1996), 731-742. doi: 10.1088/0266-5611/12/5/014. [30] R. Potthast, "Point Sources and Multipoles in Inverse Scattering Theorey," Chapman& Hall/CRC Research Notes in Math., 2001. doi: 10.1201/9781420035483. [31] R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), R1-R47. doi: 10.1088/0266-5611/22/2/R01. [32] P. Martin, "Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles," Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2006 doi: 10.1017/CBO9780511735110.
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