Figure 1.
First line : Factorization method (left) and GLSM (right) without sign changing contrast. Second line : Factorization method (left) and GLSM (right) with sign changing contrast.
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Near-field imaging of sound-soft obstacles in periodic waveguides
December
2017, 11(6): 1107-1119.
doi: 10.3934/ipi.2017051
The Generalized Linear Sampling and factorization methods only depends on the sign of contrast on the boundary
6 quai Watier, BP 49, Chatou, 78401, CEDEX, France |
We extend the applicability of the Generalized Linear Sampling Method (GLSM)[
Keywords:
Inverse scattering problems,
linear sampling method,
factorization method,
qualitative methods.
Mathematics Subject Classification:
Primary: 35R60, 35R30; Secondary: 65M32.
Citation:
Lorenzo Audibert. The Generalized Linear Sampling and factorization methods only depends on the sign of contrast on the boundary. Inverse Problems and Imaging,
2017, 11
(6)
: 1107-1119.
doi: 10.3934/ipi.2017051
![]()
References:
| [1] |
L. Audibert, A. Girard and H. Haddar,
Identifying defects in an unknown background using differential measurements, Inverse Problems and Imaging, 9 (2015), 625-643.
doi: 10.3934/ipi.2015.9.625. |
| [2] |
L. Audibert and H. Haddar, A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements, Inverse Problems, 30 (2014), 035011, 20pp.
doi: 10.1088/0266-5611/30/3/035011. |
| [3] |
L. Audibert and H. Haddar,
The generalized linear sampling method for limited aperture measurements, SIAM J. Imaging Sci., 10 (2017), 845-870.
doi: 10.1137/16M110112X. |
| [4] |
A.-S. Bonnet-Ben Dhia, L. Chesnel and H. Haddar,
On the use of $T$-coercivity to study the interior transmission eigenvalue problem, C. R. Math. Acad. Sci. Paris, 349 (2011), 647-651.
doi: 10.1016/j.crma.2011.05.008. |
| [5] |
F. Cakoni and D. Colton,
Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006. An introduction. |
| [6] |
F. Cakoni, D. Colton and H. Haddar,
Inverse Scattering Theory and Transmission Eigenvalues, volume 88 of CBMS Series publications, 2016.
doi: 10.1137/1.9781611974461.ch1. |
| [7] |
F. Cakoni and I. Harris, The factorization method for a defective region in an anisotropic material, Inverse Problems, 31 (2015), 025002, 22pp.
doi: 10.1088/0266-5611/31/2/025002. |
| [8] |
H. Haddar, F. Cakoni and S. Meng,
Boundary integral equations for the transmission eigenvalue problem for aaxwell equations, J. Integral Equations Appl., 27 (2015), 375-406.
doi: 10.1216/JIE-2015-27-3-375. |
| [9] |
A. Cossonnière and H. Haddar,
Surface integral formulation of the interior transmission problem, J. Integral Equations Appl., 25 (2013), 341-376.
doi: 10.1216/JIE-2013-25-3-341. |
| [10] |
L. Evgeny and L. Armin, Monotonicity in inverse medium scattering, |
| [11] |
B. Gebauer,
The factorization method for real elliptic problems, Zeitschrift für Analysis und ihre Anwendungen, 25 (2006), 81-102.
doi: 10.4171/ZAA/1279. |
| [12] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. |
| [13] |
P. Grisvard,
Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
| [14] |
A. Kirsch,
The factorization method for a class of inverse elliptic problems, Mathematische Nachrichten, 278 (2005), 258-277.
doi: 10.1002/mana.200310239. |
| [15] |
A. Kirsch,
A note on Sylvester's proof of discreteness of interior transmission eigenvalues, Comptes Rendus Mathématique, 354 (2016), 377-382.
doi: 10.1016/j.crma.2016.01.015. |
| [16] |
A. Kirsch and N. Grinberg,
The Factorization Method for Inverse Problems, volume 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008. |
| [17] |
J. Sylvester,
Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.
doi: 10.1137/110836420. |
| [18] |
J. Yang, B. ~Zhang and H. Zhang,
The factorization method for reconstructing a penetrable obstacle with unknown buried objects, SIAM Journal of Applied Mathematics, 73 (2013), 617-635.
doi: 10.1137/120883724. |
show all references
References:
| [1] |
L. Audibert, A. Girard and H. Haddar,
Identifying defects in an unknown background using differential measurements, Inverse Problems and Imaging, 9 (2015), 625-643.
doi: 10.3934/ipi.2015.9.625. |
| [2] |
L. Audibert and H. Haddar, A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements, Inverse Problems, 30 (2014), 035011, 20pp.
doi: 10.1088/0266-5611/30/3/035011. |
| [3] |
L. Audibert and H. Haddar,
The generalized linear sampling method for limited aperture measurements, SIAM J. Imaging Sci., 10 (2017), 845-870.
doi: 10.1137/16M110112X. |
| [4] |
A.-S. Bonnet-Ben Dhia, L. Chesnel and H. Haddar,
On the use of $T$-coercivity to study the interior transmission eigenvalue problem, C. R. Math. Acad. Sci. Paris, 349 (2011), 647-651.
doi: 10.1016/j.crma.2011.05.008. |
| [5] |
F. Cakoni and D. Colton,
Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006. An introduction. |
| [6] |
F. Cakoni, D. Colton and H. Haddar,
Inverse Scattering Theory and Transmission Eigenvalues, volume 88 of CBMS Series publications, 2016.
doi: 10.1137/1.9781611974461.ch1. |
| [7] |
F. Cakoni and I. Harris, The factorization method for a defective region in an anisotropic material, Inverse Problems, 31 (2015), 025002, 22pp.
doi: 10.1088/0266-5611/31/2/025002. |
| [8] |
H. Haddar, F. Cakoni and S. Meng,
Boundary integral equations for the transmission eigenvalue problem for aaxwell equations, J. Integral Equations Appl., 27 (2015), 375-406.
doi: 10.1216/JIE-2015-27-3-375. |
| [9] |
A. Cossonnière and H. Haddar,
Surface integral formulation of the interior transmission problem, J. Integral Equations Appl., 25 (2013), 341-376.
doi: 10.1216/JIE-2013-25-3-341. |
| [10] |
L. Evgeny and L. Armin, Monotonicity in inverse medium scattering, |
| [11] |
B. Gebauer,
The factorization method for real elliptic problems, Zeitschrift für Analysis und ihre Anwendungen, 25 (2006), 81-102.
doi: 10.4171/ZAA/1279. |
| [12] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. |
| [13] |
P. Grisvard,
Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
| [14] |
A. Kirsch,
The factorization method for a class of inverse elliptic problems, Mathematische Nachrichten, 278 (2005), 258-277.
doi: 10.1002/mana.200310239. |
| [15] |
A. Kirsch,
A note on Sylvester's proof of discreteness of interior transmission eigenvalues, Comptes Rendus Mathématique, 354 (2016), 377-382.
doi: 10.1016/j.crma.2016.01.015. |
| [16] |
A. Kirsch and N. Grinberg,
The Factorization Method for Inverse Problems, volume 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008. |
| [17] |
J. Sylvester,
Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.
doi: 10.1137/110836420. |
| [18] |
J. Yang, B. ~Zhang and H. Zhang,
The factorization method for reconstructing a penetrable obstacle with unknown buried objects, SIAM Journal of Applied Mathematics, 73 (2013), 617-635.
doi: 10.1137/120883724. |
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