American Institute of Mathematical Sciences

Advanced
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

[1]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013
[2]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123
[3]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[4]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[5]

Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems & Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263
[6]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[7]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604
[8]

Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031
[9]

Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems & Imaging, 2020, 14 (4) : 719-731. doi: 10.3934/ipi.2020033
[10]

Yunwen Yin, Weishi Yin, Pinchao Meng, Hongyu Liu. The interior inverse scattering problem for a two-layered cavity using the Bayesian method. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021069
[11]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[12]

Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems & Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017
[13]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167
[14]

Karzan Berdawood, Abdeljalil Nachaoui, Rostam Saeed, Mourad Nachaoui, Fatima Aboud. An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation. Discrete & Continuous Dynamical Systems - S, 2022, 15 (1) : 57-78. doi: 10.3934/dcdss.2021013
[15]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[16]

Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016
[17]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355
[18]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
[19]

David Colton, Lassi Päivärinta, John Sylvester. The interior transmission problem. Inverse Problems & Imaging, 2007, 1 (1) : 13-28. doi: 10.3934/ipi.2007.1.13
[20]

Fioralba Cakoni, Shixu Meng, Jingni Xiao. A note on transmission eigenvalues in electromagnetic scattering theory. Inverse Problems & Imaging, 2021, 15 (5) : 999-1014. doi: 10.3934/ipi.2021025

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy