American Institute of Mathematical Sciences

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

Imaging acoustic obstacles by singular and hypersingular point sources

Jingzhi Li 1, , Hongyu Liu 2, , Hongpeng Sun 3, and  Jun Zou 4,

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.
Keywords: hypersingular point sources., Dirichelet-to-Neumann map, sampling method, indicator function, Inverse acoustic scattering, boundary measurment.
Mathematics Subject Classification: Primary: 35R30, 35J05, 78A45; Secondary: 35P25, 81U4.
Citation: Jingzhi Li, Hongyu Liu, Hongpeng Sun, Jun Zou. Imaging acoustic obstacles by singular and hypersingular point sources. Inverse Problems & 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 (2008).   Google Scholar

[2]

H. Ammari, J. Garnier, H. Kang, M. Lim and K. Solna, Multistatic imaging of extended targets,, SIAM J. Imaging Sci., 5 (2012), 564.  doi: 10.1137/10080631X.  Google Scholar

[3]

H. Ammari, R. Griesmaier and M. Hanke, Identification of small inhomogeneities: Asymptotic factorization,, Math. Comp., 76 (2007), 1425.  doi: 10.1090/S0025-5718-07-01946-1.  Google Scholar

[4]

H. Ammari and H. Kang, "Expansion Methods,", Handbook of Mathematical Methods in Imaging, (2011), 447.  doi: 10.1090/conm/548.  Google Scholar

[5]

H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements,", Springer-Verlag, (2004).  doi: 10.1007/b98245.  Google Scholar

[6]

H. Ammari, H. Kang, H. Lee and W. K. Park, Asymptotic imaging of perfectly conducting cracks,, SIAM J. Sci. Comput., 32 (2010), 894.  doi: 10.1137/090749013.  Google Scholar

[7]

T. Arens, Why the linear sampling method works,, Inverse Problems, 20 (2004), 163.  doi: 10.1088/0266-5611/20/1/010.  Google Scholar

[8]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory,", Springer-Verlag, (2006).   Google Scholar

[9]

M. Cheney, The linear sampling method and the MUSIC algorithm,, Inverse Problems, 17 (2001), 591.  doi: 10.1088/0266-5611/17/4/301.  Google Scholar

[10]

W. C. Chew, "Waves and Fields In Inhomogenenous Media,", Van Nostrand Reinhold, (1990).   Google Scholar

[11]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic inverse scattering theory,, Inverse Problems, 19 (2003).  doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[12]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[13]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", $2^{nd}$ edition, (1998).   Google Scholar

[14]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/3/R01.  Google Scholar

[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.  doi: 10.1137/0908035.  Google Scholar

[16]

D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves,, SIAM J. Appl. Math., 58 (1998), 926.  doi: 10.1137/S0036139996308005.  Google Scholar

[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.  doi: 10.1088/0266-5611/21/6/015.  Google Scholar

[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.  doi: 10.1002/cpa.20194.  Google Scholar

[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).  doi: 10.1088/0266-5611/25/10/105005.  Google Scholar

[20]

M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data,, Inverse Problems, 15 (1999), 1231.  doi: 10.1088/0266-5611/15/5/308.  Google Scholar

[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.  doi: 10.1515/jiip.1999.7.3.255.  Google Scholar

[22]

V. Isakov, "Inverse Problems for Partial Differential Equations,", $2^{nd}$ edition, (2006).   Google Scholar

[23]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford University Press, (2008).   Google Scholar

[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.  doi: 10.1137/060674247.  Google Scholar

[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.  doi: 10.1137/080734170.  Google Scholar

[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).  doi: 10.1063/1.4751282.  Google Scholar

[27]

W. Mclean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000).   Google Scholar

[28]

G. Nakamura and K. Yoshida, Identification of a non-convex obstacle for acoustical scattering,, J. Inv. Ill-posed Problems, 15 (2007), 611.  doi: 10.1515/jiip.2007.034.  Google Scholar

[29]

R. Potthast, A fast new method to solve inverse scattering problem,, Inverse Problem, 12 (1996), 731.  doi: 10.1088/0266-5611/12/5/014.  Google Scholar

[30]

R. Potthast, "Point Sources and Multipoles in Inverse Scattering Theorey,", Chapman& Hall/CRC Research Notes in Math., (2001).  doi: 10.1201/9781420035483.  Google Scholar

[31]

R. Potthast, A survey on sampling and probe methods for inverse problems,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/2/R01.  Google Scholar

[32]

P. Martin, "Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles,", Encyclopedia of Mathematics and its Applications, (2006).  doi: 10.1017/CBO9780511735110.  Google Scholar

show all references

References:
[1]

H. Ammari, "An Introduction to Mathematics of Emerging Biomedical Imaging,", Mathematics and Applications, 62 (2008).   Google Scholar

[2]

H. Ammari, J. Garnier, H. Kang, M. Lim and K. Solna, Multistatic imaging of extended targets,, SIAM J. Imaging Sci., 5 (2012), 564.  doi: 10.1137/10080631X.  Google Scholar

[3]

H. Ammari, R. Griesmaier and M. Hanke, Identification of small inhomogeneities: Asymptotic factorization,, Math. Comp., 76 (2007), 1425.  doi: 10.1090/S0025-5718-07-01946-1.  Google Scholar

[4]

H. Ammari and H. Kang, "Expansion Methods,", Handbook of Mathematical Methods in Imaging, (2011), 447.  doi: 10.1090/conm/548.  Google Scholar

[5]

H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements,", Springer-Verlag, (2004).  doi: 10.1007/b98245.  Google Scholar

[6]

H. Ammari, H. Kang, H. Lee and W. K. Park, Asymptotic imaging of perfectly conducting cracks,, SIAM J. Sci. Comput., 32 (2010), 894.  doi: 10.1137/090749013.  Google Scholar

[7]

T. Arens, Why the linear sampling method works,, Inverse Problems, 20 (2004), 163.  doi: 10.1088/0266-5611/20/1/010.  Google Scholar

[8]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory,", Springer-Verlag, (2006).   Google Scholar

[9]

M. Cheney, The linear sampling method and the MUSIC algorithm,, Inverse Problems, 17 (2001), 591.  doi: 10.1088/0266-5611/17/4/301.  Google Scholar

[10]

W. C. Chew, "Waves and Fields In Inhomogenenous Media,", Van Nostrand Reinhold, (1990).   Google Scholar

[11]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic inverse scattering theory,, Inverse Problems, 19 (2003).  doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[12]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[13]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", $2^{nd}$ edition, (1998).   Google Scholar

[14]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/3/R01.  Google Scholar

[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.  doi: 10.1137/0908035.  Google Scholar

[16]

D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves,, SIAM J. Appl. Math., 58 (1998), 926.  doi: 10.1137/S0036139996308005.  Google Scholar

[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.  doi: 10.1088/0266-5611/21/6/015.  Google Scholar

[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.  doi: 10.1002/cpa.20194.  Google Scholar

[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).  doi: 10.1088/0266-5611/25/10/105005.  Google Scholar

[20]

M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data,, Inverse Problems, 15 (1999), 1231.  doi: 10.1088/0266-5611/15/5/308.  Google Scholar

[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.  doi: 10.1515/jiip.1999.7.3.255.  Google Scholar

[22]

V. Isakov, "Inverse Problems for Partial Differential Equations,", $2^{nd}$ edition, (2006).   Google Scholar

[23]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford University Press, (2008).   Google Scholar

[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.  doi: 10.1137/060674247.  Google Scholar

[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.  doi: 10.1137/080734170.  Google Scholar

[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).  doi: 10.1063/1.4751282.  Google Scholar

[27]

W. Mclean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000).   Google Scholar

[28]

G. Nakamura and K. Yoshida, Identification of a non-convex obstacle for acoustical scattering,, J. Inv. Ill-posed Problems, 15 (2007), 611.  doi: 10.1515/jiip.2007.034.  Google Scholar

[29]

R. Potthast, A fast new method to solve inverse scattering problem,, Inverse Problem, 12 (1996), 731.  doi: 10.1088/0266-5611/12/5/014.  Google Scholar

[30]

R. Potthast, "Point Sources and Multipoles in Inverse Scattering Theorey,", Chapman& Hall/CRC Research Notes in Math., (2001).  doi: 10.1201/9781420035483.  Google Scholar

[31]

R. Potthast, A survey on sampling and probe methods for inverse problems,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/2/R01.  Google Scholar

[32]

P. Martin, "Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles,", Encyclopedia of Mathematics and its Applications, (2006).  doi: 10.1017/CBO9780511735110.  Google Scholar

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