The Factorization Method for an inverse fluid-solid interaction scattering problem

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November 2012, 6(4): 681-695. doi: 10.3934/ipi.2012.6.681

The Factorization Method for an inverse fluid-solid interaction scattering problem

1.	Department of Mathematics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
2.	Universidad Autonoma de Madrid, Departamento de Matemáticas, Madrid, Spain

Received December 2011 Revised June 2012 Published November 2012

Abstract
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In this paper we justify the Factorization Method for a coupled acoustic-elastic medium. Under natural assumptions on the data we prove an explicit form of the characteristic function of the scattering medium $D$ where only the spectral data of the far field operator enter. This information is provided by the knowledge of the far field patterns for all incident plane waves. In the last section we investigate the corresponding interior transmission eigenvalue problem and prove that the eigenvalues form a discrete set.

Keywords: Factorization Method, Inverse scattering problem, Lamé's equation, interior transmission eigenvalues., Helmholtz equation.

Mathematics Subject Classification: Primary: 35P25, 35R30; Secondary: 35P9.

Citation: Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

References:

[1]	G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral obstacle by a single far-field measurement,, Proc. Am. Math. Soc., 6 (2005), 1685. doi: 10.1090/S0002-9939-05-07810-X.
[2]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory: An Introduction,", Interaction of Mechanics and Mathematics, (2006).
[3]	F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math Anal., 42 (2010), 237. doi: 10.1137/090769338.
[4]	F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium,, Applicable Analysis, 88 (2009), 475. doi: 10.1080/00036810802713966.
[5]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", 2nd edition, 93 (1998).
[6]	D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13. doi: 10.3934/ipi.2007.1.13.
[7]	G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solutions of fluid-solid interaction problems,, Math. Nachr., 218 (2000), 139. doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S.
[8]	A. Kirsch, On the existence of transmission eigenvalues,, Inverse Problems and Imaging, 3 (2009), 155. doi: 10.3934/ipi.2009.3.155.
[9]	A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, 36 (2008).
[10]	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. doi: 10.1088/0266-5611/22/2/008.
[11]	C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle,, SIAM J. Appl. Math., 55 (1995), 904. doi: 10.1137/S0036139993259027.
[12]	, P. Monk,, Personal Communication, (2012).
[13]	P. Monk and V. Selgas, An inverse fluid-solid interaction problem,, Inverse Problems and Imaging, 3 (2009), 173. doi: 10.3934/ipi.2009.3.173.
[14]	P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem,, Inverse Problems and Imaging, 5 (2011), 465. doi: 10.3934/ipi.2011.5.465.
[15]	D. Natroshvili, S. Kharibegashvili and Z. Tediasvili, Direct and inverse fluid-structure interaction problems,, Rendiconti di Matematica, 20 (2000), 57.
[16]	L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738.

show all references

References:

[1]	G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral obstacle by a single far-field measurement,, Proc. Am. Math. Soc., 6 (2005), 1685. doi: 10.1090/S0002-9939-05-07810-X.
[2]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory: An Introduction,", Interaction of Mechanics and Mathematics, (2006).
[3]	F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math Anal., 42 (2010), 237. doi: 10.1137/090769338.
[4]	F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium,, Applicable Analysis, 88 (2009), 475. doi: 10.1080/00036810802713966.
[5]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", 2nd edition, 93 (1998).
[6]	D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13. doi: 10.3934/ipi.2007.1.13.
[7]	G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solutions of fluid-solid interaction problems,, Math. Nachr., 218 (2000), 139. doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S.
[8]	A. Kirsch, On the existence of transmission eigenvalues,, Inverse Problems and Imaging, 3 (2009), 155. doi: 10.3934/ipi.2009.3.155.
[9]	A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, 36 (2008).
[10]	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. doi: 10.1088/0266-5611/22/2/008.
[11]	C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle,, SIAM J. Appl. Math., 55 (1995), 904. doi: 10.1137/S0036139993259027.
[12]	, P. Monk,, Personal Communication, (2012).
[13]	P. Monk and V. Selgas, An inverse fluid-solid interaction problem,, Inverse Problems and Imaging, 3 (2009), 173. doi: 10.3934/ipi.2009.3.173.
[14]	P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem,, Inverse Problems and Imaging, 5 (2011), 465. doi: 10.3934/ipi.2011.5.465.
[15]	D. Natroshvili, S. Kharibegashvili and Z. Tediasvili, Direct and inverse fluid-structure interaction problems,, Rendiconti di Matematica, 20 (2000), 57.
[16]	L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738.

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