The factorization method for scatterers with different physical properties

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June 2015, 8(3): 563-577. doi: 10.3934/dcdss.2015.8.563

The factorization method for scatterers with different physical properties

Xiaodong Liu ^1,

1.	Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Received November 2013 Revised January 2014 Published October 2014

Abstract
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The scattering of time-harmonic acoustic plane waves by a mixed type scatterer is considered. Such a scatterer is given as the union of several components with different physical properties. Some of them are impenetrable obstacles with Dirichlet or impedance boundary conditions, while the others are penetrable inhomogeneous media with compact support. This paper is concerned with modifications of the factorization method for the following two basic cases, one is the scattering by a priori separated sound-soft and sound-hard obstacles, the other one is the scattering by a scatterer with impenetrable (Dirichlet) and penetrable components. The other cases can be dealt with similarly. Finally, some numerical experiments are presented to demonstrate the feasibility and effectiveness of the modified factorization methods.

Keywords: factorization method, inhomogeneous medium., Acoustic scattering, impenetrable obstacle, mixed type scatterers.

Mathematics Subject Classification: Primary: 35P25, 35R30; Secondary: 45Q05, 78A4.

Citation: Xiaodong Liu. The factorization method for scatterers with different physical properties. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 563-577. doi: 10.3934/dcdss.2015.8.563

References:

[1]	O. Bondarenko, A. Kirsch and X. Liu, The Factorization method for inverse acoustic scattering in a layered medium, Inverse Problems, 29 (2013), 045010. doi: 10.1088/0266-5611/29/4/045010.
[2]	O. Bondarenko and X. Liu, The Factorization method for inverse obstacle scattering with conductive boundary condition, Inverse Problems, 29 (2013), 095021. doi: 10.1088/0266-5611/29/9/095021.
[3]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd Edition, Springer, 2012. doi: 10.1007/978-1-4614-4942-3.
[4]	N. Grinberg and A. Kirsch, The Linear sampling method in inverse obstacle scattering for impedance boundary conditions, Journal of Inverse and Ill-Posed Problems, 10 (2002), 171-185. doi: 10.1515/jiip.2002.10.2.171.
[5]	N. Grinberg and A. Kirsch, The factorization method for obstacles with a-prior separated sound-soft and sound-hard parts, Math. Comput. in Simul., 66, (2004), 267-279. doi: 10.1016/j.matcom.2004.02.011.
[6]	A. Kirsch, Charaterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.
[7]	A. Kirsch, Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, 15 (1999), 413-429. doi: 10.1088/0266-5611/15/2/005.
[8]	A. Kirsch, The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.
[9]	A. Kirsch, An introduction to the Mathematical Theory of Inverse Problems, 2nd Edition, Springer, 2011. doi: 10.1007/978-1-4419-8474-6.
[10]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, 2008.
[11]	A. Kirsch and X. Liu, The factorization method for inverse acoustic scattering by a penetrable anisotropic obstacle, Math. Meth. Appl. Sci., 37 (2014), 1159-1170. doi: 10.1002/mma.2877.
[12]	A. Kirsch and X. Liu, Direct and inverse acoustic scattering by a mixed-type scatterer, Inverse Problems, 29 (2013), 065005. doi: 10.1088/0266-5611/29/6/065005.
[13]	A. Kirsch and X. Liu, A modification of the factorization method for the classical acoustic inverse scattering problems, Inverse Problems, 30 (2014), 035013. doi: 10.1088/0266-5611/30/3/035013.
[14]	A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Problems Imaging, 3 (2009), 123-138. doi: 10.3934/ipi.2009.3.123.
[15]	X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006. doi: 10.1088/0266-5611/30/1/015006.
[16]	W. Mclean, Strongly Elliptic Systems and Boundary Integral Equation, Cambridge University Press, Cambridge, 2000.
[17]	D. L. Nguyen, Spectral Methods for Direct and Inverse Scattering from Periodic Structures, PhD thesis, Ecole Polytechnique X, Paris, 2012.
[18]	J. Yang, B. Zhang and H. Zhang, The factorization method for reconstructing a penetrable obstacle with unknown buried objects, SIAM J. Appl. Math., 73 (2013), 617-635. doi: 10.1137/120883724.

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References:

[1]	O. Bondarenko, A. Kirsch and X. Liu, The Factorization method for inverse acoustic scattering in a layered medium, Inverse Problems, 29 (2013), 045010. doi: 10.1088/0266-5611/29/4/045010.
[2]	O. Bondarenko and X. Liu, The Factorization method for inverse obstacle scattering with conductive boundary condition, Inverse Problems, 29 (2013), 095021. doi: 10.1088/0266-5611/29/9/095021.
[3]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd Edition, Springer, 2012. doi: 10.1007/978-1-4614-4942-3.
[4]	N. Grinberg and A. Kirsch, The Linear sampling method in inverse obstacle scattering for impedance boundary conditions, Journal of Inverse and Ill-Posed Problems, 10 (2002), 171-185. doi: 10.1515/jiip.2002.10.2.171.
[5]	N. Grinberg and A. Kirsch, The factorization method for obstacles with a-prior separated sound-soft and sound-hard parts, Math. Comput. in Simul., 66, (2004), 267-279. doi: 10.1016/j.matcom.2004.02.011.
[6]	A. Kirsch, Charaterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.
[7]	A. Kirsch, Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory, Inverse Problems, 15 (1999), 413-429. doi: 10.1088/0266-5611/15/2/005.
[8]	A. Kirsch, The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.
[9]	A. Kirsch, An introduction to the Mathematical Theory of Inverse Problems, 2nd Edition, Springer, 2011. doi: 10.1007/978-1-4419-8474-6.
[10]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, 2008.
[11]	A. Kirsch and X. Liu, The factorization method for inverse acoustic scattering by a penetrable anisotropic obstacle, Math. Meth. Appl. Sci., 37 (2014), 1159-1170. doi: 10.1002/mma.2877.
[12]	A. Kirsch and X. Liu, Direct and inverse acoustic scattering by a mixed-type scatterer, Inverse Problems, 29 (2013), 065005. doi: 10.1088/0266-5611/29/6/065005.
[13]	A. Kirsch and X. Liu, A modification of the factorization method for the classical acoustic inverse scattering problems, Inverse Problems, 30 (2014), 035013. doi: 10.1088/0266-5611/30/3/035013.
[14]	A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Problems Imaging, 3 (2009), 123-138. doi: 10.3934/ipi.2009.3.123.
[15]	X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006. doi: 10.1088/0266-5611/30/1/015006.
[16]	W. Mclean, Strongly Elliptic Systems and Boundary Integral Equation, Cambridge University Press, Cambridge, 2000.
[17]	D. L. Nguyen, Spectral Methods for Direct and Inverse Scattering from Periodic Structures, PhD thesis, Ecole Polytechnique X, Paris, 2012.
[18]	J. Yang, B. Zhang and H. Zhang, The factorization method for reconstructing a penetrable obstacle with unknown buried objects, SIAM J. Appl. Math., 73 (2013), 617-635. doi: 10.1137/120883724.

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