# American Institute of Mathematical Sciences

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

Received  February 2017 Published  September 2017

We extend the applicability of the Generalized Linear Sampling Method (GLSM)[2] and the Factorization Method (FM)[16] to the case of inhomogeneities where the contrast changes sign. Both methods give an exact characterization of the target shapes in terms of the farfield operator (at a fixed frequency) using the coercivity property of a special solution operator. We prove this property assuming that the contrast has a fixed sign in a neighborhood of the inhomogeneities boundary. We treat both isotropic and anisotropic scatterers with possibly different supports for the isotropic and anisotropic parts. We finally validate the methods through some numerical tests in two dimensions.

Citation: Lorenzo Audibert. The Generalized Linear Sampling and factorization methods only depends on the sign of contrast on the boundary. Inverse Problems & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006. An introduction.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] L. Evgeny and L. Armin, Monotonicity in inverse medium scattering, Google Scholar [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.  Google Scholar [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.  Google Scholar [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar [14] A. Kirsch, The factorization method for a class of inverse elliptic problems, Mathematische Nachrichten, 278 (2005), 258-277.  doi: 10.1002/mana.200310239.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.  doi: 10.1137/110836420.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006. An introduction.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] L. Evgeny and L. Armin, Monotonicity in inverse medium scattering, Google Scholar [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.  Google Scholar [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.  Google Scholar [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar [14] A. Kirsch, The factorization method for a class of inverse elliptic problems, Mathematische Nachrichten, 278 (2005), 258-277.  doi: 10.1002/mana.200310239.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.  doi: 10.1137/110836420.  Google Scholar [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.  Google Scholar
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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