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A three-dimensional inverse gravimetry problem for ice with snow caps
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 |
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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