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Error bounds and stability in the $l_{0}$ regularized for CT reconstruction from small projections
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 |
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. |
show all references
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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