American Institute of Mathematical Sciences

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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

Andreas Kirsch 1, and  Albert Ruiz 2,

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

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), 1685-1691. doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar

[2]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory: An Introduction," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.  Google Scholar

[3]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math Anal., 42 (2010), 237-255. doi: 10.1137/090769338.  Google Scholar

[4]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Applicable Analysis, 88 (2009), 475-493. doi: 10.1080/00036810802713966.  Google Scholar

[5]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.  Google Scholar

[6]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Problems and Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13.  Google Scholar

[7]

G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., 218 (2000), 139-163. doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S.  Google Scholar

[8]

A. Kirsch, On the existence of transmission eigenvalues, Inverse Problems and Imaging, 3 (2009), 155-172. doi: 10.3934/ipi.2009.3.155.  Google Scholar

[9]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.  Google Scholar

[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), 515-524. doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[11]

C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math., 55 (1995), 904-922. doi: 10.1137/S0036139993259027.  Google Scholar

[12]

, P. Monk,, Personal Communication, (2012).   Google Scholar

[13]

P. Monk and V. Selgas, An inverse fluid-solid interaction problem, Inverse Problems and Imaging, 3 (2009), 173-198. doi: 10.3934/ipi.2009.3.173.  Google Scholar

[14]

P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem, Inverse Problems and Imaging, 5 (2011), 465-483. doi: 10.3934/ipi.2011.5.465.  Google Scholar

[15]

D. Natroshvili, S. Kharibegashvili and Z. Tediasvili, Direct and inverse fluid-structure interaction problems, Rendiconti di Matematica, Serie VII, 20 (2000), 57-92.  Google Scholar

[16]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753.  Google Scholar

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), 1685-1691. doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar

[2]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory: An Introduction," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.  Google Scholar

[3]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math Anal., 42 (2010), 237-255. doi: 10.1137/090769338.  Google Scholar

[4]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Applicable Analysis, 88 (2009), 475-493. doi: 10.1080/00036810802713966.  Google Scholar

[5]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.  Google Scholar

[6]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Problems and Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13.  Google Scholar

[7]

G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., 218 (2000), 139-163. doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S.  Google Scholar

[8]

A. Kirsch, On the existence of transmission eigenvalues, Inverse Problems and Imaging, 3 (2009), 155-172. doi: 10.3934/ipi.2009.3.155.  Google Scholar

[9]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.  Google Scholar

[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), 515-524. doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[11]

C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math., 55 (1995), 904-922. doi: 10.1137/S0036139993259027.  Google Scholar

[12]

, P. Monk,, Personal Communication, (2012).   Google Scholar

[13]

P. Monk and V. Selgas, An inverse fluid-solid interaction problem, Inverse Problems and Imaging, 3 (2009), 173-198. doi: 10.3934/ipi.2009.3.173.  Google Scholar

[14]

P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem, Inverse Problems and Imaging, 5 (2011), 465-483. doi: 10.3934/ipi.2011.5.465.  Google Scholar

[15]

D. Natroshvili, S. Kharibegashvili and Z. Tediasvili, Direct and inverse fluid-structure interaction problems, Rendiconti di Matematica, Serie VII, 20 (2000), 57-92.  Google Scholar

[16]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753.  Google Scholar

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