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A direct sampling method for inverse scattering using far-field data
1. | Faculty of Science, South University of Science and Technology of China, Shenzhen, 518055 |
2. | Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong |
References:
[1] |
M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions," Dover, NewYork, 1965.
doi: 10.1119/1.1972842. |
[2] |
G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Am. Math., 133 (2005), 1685-1691.
doi: 10.1090/S0002-9939-05-07810-X. |
[3] |
H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements," Lecture Notes in Mathematics, 1846, Springer-Verlag, Berlin, 2004.
doi: 10.1007/b98245. |
[4] |
G. Bao and P. Li, Inverse medium scattering for the Helmholtz equation at fixed frequency, Inverse Problems, 21 (2005), 1621-1641.
doi: 10.1088/0266-5611/21/5/007. |
[5] |
L. Beilina and M. V. Klibanov, "Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems," Springer, New York, 2012.
doi: 10.1007/978-1-4419-7805-9. |
[6] |
J. Buchanan, R. Gilbert, A. Wirgin and Y. Xu, "Marine Acoustics: Direct and Inverse Scattering of Waves," Society for Industrial and Applied Mathematics, Philadelphia, PA, 2004.
doi: 10.1137/1.9780898717983. |
[7] |
F. Cakoni, D. Colton and P. Monk, "The Linear Sampling Method in Inverse Electromagnetic Scattering," CBMS-NSF Regional Conference Series in Applied Mathematics, 80. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
doi: 10.1137/1.9780898719406. |
[8] |
X. Chen and Y. Zhong, Music electromagnetic imaging with enhanced resolution for small inclusions, Inverse Problems, 25 (2009), 015008 (12pp).
doi: 10.1088/0266-5611/25/1/015008. |
[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] |
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. |
[11] |
D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983. |
[12] |
D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition. Applied Mathematical Sciences, 93. Springer-Verlag, Berlin, 1998. |
[13] |
D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA Journal of Applied Mathematics, 31 (1983), 253-259.
doi: 10.1093/imamat/31.3.253. |
[14] |
A. Devaney, Super-resolution processing of multi-static data using time-reversal and music,, J. Acoust. Soc. Am., ().
|
[15] |
J. Elschner and M. Yamamoto, Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave, Inverse Problems, 24 (2008), 035004 (7pp).
doi: 10.1088/0266-5611/24/3/035004. |
[16] |
R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Problems, 27 (2011), 085005 (23pp).
doi: 10.1088/0266-5611/27/8/085005. |
[17] |
F. K. Gruber, E. A. Marengo and A. J. Devaney, Time-reversal imaging with multiple signal classification considering multiple scattering between the targets, J. Acoust. Soc. Am., 115 (2004), 3042-3047.
doi: 10.1121/1.1738451. |
[18] |
M. Hanke, One shot inverse scattering via rational approximation, SIAM J. Imaging Sciences, 5 (2012), 465-482.
doi: 10.1137/110823985. |
[19] |
T. Hohage, On the numerical solution of a three-dimensional inverse medium scattering problem, Inverse Problems, 17 (2001), 1743-1763.
doi: 10.1088/0266-5611/17/6/314. |
[20] |
S. Hou, K. Solna and H. Zhao, A direct imaging algorithm for extended targets, Inverse Problems, 22 (2006), 1151-1178.
doi: 10.1088/0266-5611/22/4/003. |
[21] |
K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003 (11pp).
doi: 10.1088/0266-5611/28/2/025003. |
[22] |
K. Ito, B. Jin and J. Zou, A two-stage method for inverse medium scattering, J. Comput. Phys., 237 (2013), 211-223.
doi: 10.1016/j.jcp.2012.12.004. |
[23] |
A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1511.
doi: 10.1088/0266-5611/14/6/009. |
[24] |
A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36. Oxford University Press, Oxford, 2008. |
[25] |
R. Kohn, D. Onofrei, M. Vogelius and M. Weinstein, Cloaking via change of variables for the helmholtz equation, Comm. Pure Appl. Math., 63 (2010), 973-1016.
doi: 10.1002/cpa.20326. |
[26] |
J. Li, H. Liu and H. Sun, Enhanced approximate cloaking by SH and FSH lining, Inverse Problems, 28 (2012), 075011 (21pp).
doi: 10.1088/0266-5611/28/7/075011. |
[27] |
H. Liu and H. Sun, Enhanced near-cloak by FSH lining, J. Math. Pures Appl. 99 (2013), 17-42.
doi: 10.1016/j.matpur.2012.06.001. |
[28] |
H. Liu, M. Yamamoto and J. Zou, Reflection principle for the maxwell equations and its application to inverse electromagnetic scattering, Inverse Problems, 23 (2007), 2357-2366.
doi: 10.1088/0266-5611/23/6/005. |
[29] |
H. Liu, H. Zhang and J. Zou, Recovery of polyhedral scatterers by a single electromagnetic far-field measurement, J. Math. Phy., 50 (2009), 123506 (10pp).
doi: 10.1063/1.3263140. |
[30] |
H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.
doi: 10.1088/0266-5611/22/2/008. |
[31] |
C. Müller, "Analysis of Spherial Symmetries in Eulidean Spaes," Springer, New York, 1997. |
[32] |
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. |
[33] |
R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015 (17pp).
doi: 10.1088/0266-5611/26/7/074015. |
[34] |
R. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag., 34 (1986), 276-280.
doi: 10.1109/TAP.1986.1143830. |
[35] |
C. G. Someda, "Electromagnetic Waves," Boca Raton, FL: CRC Press, 2 ed., 2006. |
[36] |
P. M. van den Berg, A. L. van Broekhoven and A. Abubakar, Extended contrast source inversion, Inverse Problems, 15 (1999), 1325-1344.
doi: 10.1088/0266-5611/15/5/315. |
[37] |
G. N. Watson, "A Treatise on the Theory of Bessel Functions," Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. vi+804 pp. |
show all references
References:
[1] |
M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions," Dover, NewYork, 1965.
doi: 10.1119/1.1972842. |
[2] |
G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Am. Math., 133 (2005), 1685-1691.
doi: 10.1090/S0002-9939-05-07810-X. |
[3] |
H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements," Lecture Notes in Mathematics, 1846, Springer-Verlag, Berlin, 2004.
doi: 10.1007/b98245. |
[4] |
G. Bao and P. Li, Inverse medium scattering for the Helmholtz equation at fixed frequency, Inverse Problems, 21 (2005), 1621-1641.
doi: 10.1088/0266-5611/21/5/007. |
[5] |
L. Beilina and M. V. Klibanov, "Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems," Springer, New York, 2012.
doi: 10.1007/978-1-4419-7805-9. |
[6] |
J. Buchanan, R. Gilbert, A. Wirgin and Y. Xu, "Marine Acoustics: Direct and Inverse Scattering of Waves," Society for Industrial and Applied Mathematics, Philadelphia, PA, 2004.
doi: 10.1137/1.9780898717983. |
[7] |
F. Cakoni, D. Colton and P. Monk, "The Linear Sampling Method in Inverse Electromagnetic Scattering," CBMS-NSF Regional Conference Series in Applied Mathematics, 80. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
doi: 10.1137/1.9780898719406. |
[8] |
X. Chen and Y. Zhong, Music electromagnetic imaging with enhanced resolution for small inclusions, Inverse Problems, 25 (2009), 015008 (12pp).
doi: 10.1088/0266-5611/25/1/015008. |
[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] |
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. |
[11] |
D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983. |
[12] |
D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition. Applied Mathematical Sciences, 93. Springer-Verlag, Berlin, 1998. |
[13] |
D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA Journal of Applied Mathematics, 31 (1983), 253-259.
doi: 10.1093/imamat/31.3.253. |
[14] |
A. Devaney, Super-resolution processing of multi-static data using time-reversal and music,, J. Acoust. Soc. Am., ().
|
[15] |
J. Elschner and M. Yamamoto, Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave, Inverse Problems, 24 (2008), 035004 (7pp).
doi: 10.1088/0266-5611/24/3/035004. |
[16] |
R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Problems, 27 (2011), 085005 (23pp).
doi: 10.1088/0266-5611/27/8/085005. |
[17] |
F. K. Gruber, E. A. Marengo and A. J. Devaney, Time-reversal imaging with multiple signal classification considering multiple scattering between the targets, J. Acoust. Soc. Am., 115 (2004), 3042-3047.
doi: 10.1121/1.1738451. |
[18] |
M. Hanke, One shot inverse scattering via rational approximation, SIAM J. Imaging Sciences, 5 (2012), 465-482.
doi: 10.1137/110823985. |
[19] |
T. Hohage, On the numerical solution of a three-dimensional inverse medium scattering problem, Inverse Problems, 17 (2001), 1743-1763.
doi: 10.1088/0266-5611/17/6/314. |
[20] |
S. Hou, K. Solna and H. Zhao, A direct imaging algorithm for extended targets, Inverse Problems, 22 (2006), 1151-1178.
doi: 10.1088/0266-5611/22/4/003. |
[21] |
K. Ito, B. Jin and J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003 (11pp).
doi: 10.1088/0266-5611/28/2/025003. |
[22] |
K. Ito, B. Jin and J. Zou, A two-stage method for inverse medium scattering, J. Comput. Phys., 237 (2013), 211-223.
doi: 10.1016/j.jcp.2012.12.004. |
[23] |
A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1511.
doi: 10.1088/0266-5611/14/6/009. |
[24] |
A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36. Oxford University Press, Oxford, 2008. |
[25] |
R. Kohn, D. Onofrei, M. Vogelius and M. Weinstein, Cloaking via change of variables for the helmholtz equation, Comm. Pure Appl. Math., 63 (2010), 973-1016.
doi: 10.1002/cpa.20326. |
[26] |
J. Li, H. Liu and H. Sun, Enhanced approximate cloaking by SH and FSH lining, Inverse Problems, 28 (2012), 075011 (21pp).
doi: 10.1088/0266-5611/28/7/075011. |
[27] |
H. Liu and H. Sun, Enhanced near-cloak by FSH lining, J. Math. Pures Appl. 99 (2013), 17-42.
doi: 10.1016/j.matpur.2012.06.001. |
[28] |
H. Liu, M. Yamamoto and J. Zou, Reflection principle for the maxwell equations and its application to inverse electromagnetic scattering, Inverse Problems, 23 (2007), 2357-2366.
doi: 10.1088/0266-5611/23/6/005. |
[29] |
H. Liu, H. Zhang and J. Zou, Recovery of polyhedral scatterers by a single electromagnetic far-field measurement, J. Math. Phy., 50 (2009), 123506 (10pp).
doi: 10.1063/1.3263140. |
[30] |
H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.
doi: 10.1088/0266-5611/22/2/008. |
[31] |
C. Müller, "Analysis of Spherial Symmetries in Eulidean Spaes," Springer, New York, 1997. |
[32] |
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. |
[33] |
R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015 (17pp).
doi: 10.1088/0266-5611/26/7/074015. |
[34] |
R. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag., 34 (1986), 276-280.
doi: 10.1109/TAP.1986.1143830. |
[35] |
C. G. Someda, "Electromagnetic Waves," Boca Raton, FL: CRC Press, 2 ed., 2006. |
[36] |
P. M. van den Berg, A. L. van Broekhoven and A. Abubakar, Extended contrast source inversion, Inverse Problems, 15 (1999), 1325-1344.
doi: 10.1088/0266-5611/15/5/315. |
[37] |
G. N. Watson, "A Treatise on the Theory of Bessel Functions," Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. vi+804 pp. |
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