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Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials
The factorization method applied to cracks with impedance boundary conditions
1. | INRIA Saclay Ile de France/Ecole Polytechnique, CMAP, Route de Saclay, 91128 Palaiseau Cedex, France |
2. | INRIA Saclay Ile de France / CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex |
References:
[1] |
G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proceedings of the American Mathematical Society, 133 (2005), 1685-1691, (electronic).
doi: 10.1090/S0002-9939-05-07810-X. |
[2] |
H. Ammari, J. Garnier, H. Kang, W. K. Park and K. Solna, Imaging schemes for perfectly conducting cracks, SIAM, J. Appl. Math., 71 (2011), 68-91.
doi: 10.1137/100800130. |
[3] |
A. Ben Abda, F. Delbary and H. Haddar, On the use of the reciprocity-gap functional in inverse scattering from planar cracks, Math. Models Methods Appl. Sci.,15 (2005), 1553-1574.
doi: 10.1142/S0218202505000819. |
[4] |
F. Ben Hassen, Y. Boukari and H. Haddar, Application of the linear sampling method to retrieve cracks with impedance boundary conditions, Inverse Problems in Science and Engineering, 2012.
doi: 10.1080/17415977.2012.686997. |
[5] |
M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem. 35 (2011), 223-235.
doi: 10.1016/j.enganabound.2010.08.007. |
[6] |
M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal. 35 (2001), 595-605.
doi: 10.1051/m2an:2001128. |
[7] |
K. Bryan and M. S. Vogelius, A review of selected works on crack identification, in Geometric Methods in Inverse Problems and PDE Control (Springer, New York), MA Vol. 137, Math. Appl., 2004, 25-46.
doi: 10.1007/978-1-4684-9375-7_3. |
[8] |
F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, 2006. |
[9] |
F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295.
doi: 10.1088/0266-5611/19/2/303. |
[10] |
N. Zeev and F. Cakoni, The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 69 (2009), 1024-1042.
doi: 10.1137/070711542. |
[11] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second edition, Heidelberg: Springer, 1998. |
[12] |
D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math., 200 (2007), 21-31.
doi: 10.1016/j.cam.2005.12.004. |
[13] |
T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern, IMA Journal of Applied Mathematics, 72 (2007), 96-112.
doi: 10.1093/imamat/hxl026. |
[14] |
O. Ivanyshyn and R. Kress, Inverse scattering for planar cracks via nonlinear integral equations, Math. Methods Appl. Sci., 31 (2008), 1221-1232.
doi: 10.1002/mma.970. |
[15] |
N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions, J. Inverse Ill-Posed Probl., 10 (2002), 171-185.
doi: 10.1515/jiip.2002.10.2.171. |
[16] |
A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture in Mathematics and Its Applications 36, (2008). |
[17] |
A. Kirsch and S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Problems, 16 (2000), 89-105.
doi: 10.1088/0266-5611/16/1/308. |
[18] |
R. Kress and P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems, 21 (2005), 773-784.
doi: 10.1088/0266-5611/21/2/020. |
[19] |
J. J. Liu, P. A. Krutitskii and M. Sini, Numerical solution of the scattering problem for acoustic waves by a two-sided crack in 2-dimensional space, J. Comput. Math., 29 (2011), 141-166.
doi: 10.4208/jams.012111.012811a. |
[20] |
A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 123-138.
doi: 10.3934/ipi.2009.3.123. |
[21] |
J. Liu and M. Sini, Reconstruction of cracks of different types from far-field measurements, Math. Meth. Appl. Sci., 33 (2010), 950-973.
doi: 10.1002/mma.1203. |
[22] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. |
[23] |
J. C. Nédélec, Acoustic and Electromagnetic Equations, Applied Matimatical Sciences. Springer-Verlag, Berlin, 2001. |
show all references
References:
[1] |
G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proceedings of the American Mathematical Society, 133 (2005), 1685-1691, (electronic).
doi: 10.1090/S0002-9939-05-07810-X. |
[2] |
H. Ammari, J. Garnier, H. Kang, W. K. Park and K. Solna, Imaging schemes for perfectly conducting cracks, SIAM, J. Appl. Math., 71 (2011), 68-91.
doi: 10.1137/100800130. |
[3] |
A. Ben Abda, F. Delbary and H. Haddar, On the use of the reciprocity-gap functional in inverse scattering from planar cracks, Math. Models Methods Appl. Sci.,15 (2005), 1553-1574.
doi: 10.1142/S0218202505000819. |
[4] |
F. Ben Hassen, Y. Boukari and H. Haddar, Application of the linear sampling method to retrieve cracks with impedance boundary conditions, Inverse Problems in Science and Engineering, 2012.
doi: 10.1080/17415977.2012.686997. |
[5] |
M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem. 35 (2011), 223-235.
doi: 10.1016/j.enganabound.2010.08.007. |
[6] |
M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal. 35 (2001), 595-605.
doi: 10.1051/m2an:2001128. |
[7] |
K. Bryan and M. S. Vogelius, A review of selected works on crack identification, in Geometric Methods in Inverse Problems and PDE Control (Springer, New York), MA Vol. 137, Math. Appl., 2004, 25-46.
doi: 10.1007/978-1-4684-9375-7_3. |
[8] |
F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, 2006. |
[9] |
F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295.
doi: 10.1088/0266-5611/19/2/303. |
[10] |
N. Zeev and F. Cakoni, The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 69 (2009), 1024-1042.
doi: 10.1137/070711542. |
[11] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second edition, Heidelberg: Springer, 1998. |
[12] |
D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math., 200 (2007), 21-31.
doi: 10.1016/j.cam.2005.12.004. |
[13] |
T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern, IMA Journal of Applied Mathematics, 72 (2007), 96-112.
doi: 10.1093/imamat/hxl026. |
[14] |
O. Ivanyshyn and R. Kress, Inverse scattering for planar cracks via nonlinear integral equations, Math. Methods Appl. Sci., 31 (2008), 1221-1232.
doi: 10.1002/mma.970. |
[15] |
N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions, J. Inverse Ill-Posed Probl., 10 (2002), 171-185.
doi: 10.1515/jiip.2002.10.2.171. |
[16] |
A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture in Mathematics and Its Applications 36, (2008). |
[17] |
A. Kirsch and S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Problems, 16 (2000), 89-105.
doi: 10.1088/0266-5611/16/1/308. |
[18] |
R. Kress and P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems, 21 (2005), 773-784.
doi: 10.1088/0266-5611/21/2/020. |
[19] |
J. J. Liu, P. A. Krutitskii and M. Sini, Numerical solution of the scattering problem for acoustic waves by a two-sided crack in 2-dimensional space, J. Comput. Math., 29 (2011), 141-166.
doi: 10.4208/jams.012111.012811a. |
[20] |
A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 123-138.
doi: 10.3934/ipi.2009.3.123. |
[21] |
J. Liu and M. Sini, Reconstruction of cracks of different types from far-field measurements, Math. Meth. Appl. Sci., 33 (2010), 950-973.
doi: 10.1002/mma.1203. |
[22] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. |
[23] |
J. C. Nédélec, Acoustic and Electromagnetic Equations, Applied Matimatical Sciences. Springer-Verlag, Berlin, 2001. |
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