Near field sampling type methods for the inverse fluid--solid interaction problem

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May 2011, 5(2): 465-483. doi: 10.3934/ipi.2011.5.465

Near field sampling type methods for the inverse fluid--solid interaction problem

Peter Monk ^1, and Virginia Selgas ^2,

1.	Department of Mathematical Sciences, University of Delaware, Newark, DE 19716
2.	Departamento de Matemáticas, Universidad de A Coruña, 15707 A Coruña

Received April 2010 Revised July 2010 Published May 2011

Abstract
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The inverse fluid--solid interaction problem considered here is to determine the shape of an elastic body from pressure measurements made in the near field. In particular we assume that the elastic body is probed by pressure waves due to point sources, and the resulting scattered field and the normal derivative of the scattered field is available for every source and receiver combination on the source and measurement curves. We provide an analysis of the Reciprocity Gap (RG) method in this case, as well as the Linear Sampling Method (LSM). A novelty of our analysis is that we exhibit a connection between the RG method and a non--standard LSM using sources and receivers on different curves. We provide numerical tests of the algorithms using both synthetic and real data.

Keywords: Linear Sampling Method, fluid--solid interaction, inverse scattering, near field..

Mathematics Subject Classification: 65N21, 74F1.

Citation: Peter Monk, Virginia Selgas. Near field sampling type methods for the inverse fluid--solid interaction problem. Inverse Problems and Imaging, 2011, 5 (2) : 465-483. doi: 10.3934/ipi.2011.5.465

References:

[1]	F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects, Inv. Prob., 22 (2006), 845-867.
[2]	F. Cakoni and H. Haddar, "A New Linear Sampling Method for the Electromagnetic Imagining of Buried Objects," in Mathematical methods in scattering theory and biomedical engineering, World Sci. Publ., Hackensack, NJ, 2006, 19-30. doi: 10.1142/9789812773197_0003.
[3]	D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Rev., 42 (2000), 369-414. doi: 10.1137/S0036144500367337.
[4]	D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inv. Prob., 21 (2005), 383-398.
[5]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2^nd edition, Springer-Verlag, New York, 1998.
[6]	D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inv. Prob., 13 (1997), 1477-1493.
[7]	J. Elschner, G. Hsiao and A. Rathsfeld, An inverse problem for fluid-solid interaction, Inverse Problems and Imaging, 2 (2007), 83-119.
[8]	J. Elschner, G. Hsiao and A. Rathsfeld, An optimization method in inverse acoustic scattering by an elastic obstacle, SIAM J. Appl. Math., 70 (2009), 168-187. doi: 10.1137/080736922.
[9]	J. Elschner, G. Hsiao and A. Rathsfeld, Comparison of numerical methods for the reconstruction of elastic obstacles from the far-field data of scattered acoustic waves, WAIS preprint No. 1479, 2010
[10]	T. Hargé, Valeurs propres d'un corps élastique, C. R. Acad. Sci. Paris, Sér. I Math., 311 (1990), 857-859.
[11]	G. Hsiao, R. Kleinman and G. F.Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., (2000), 139-163.
[12]	T. Huttunen, J. Kaipio and P. Monk, An ultra-weak method for acoustic fluid-solid interaction, J. Comput. Appl. Math., 213 (2008), 166-185. doi: 10.1016/j.cam.2006.12.030.
[13]	A. Kirsch and R. Kress, An optimization method in inverse acoustic scattering, in Boundary Elements IX (eds. C. Brebbia, W. Wendland and G. Kuhn), Springer, Heidelberg, 1987, 3-18.
[14]	C. 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.
[15]	A. Márquez, S. Meddahi and V. Selgas, A new BEM-FEM coupling strategy for two-dimensional fluid-solid interaction problems, J. Comput. Phys., 199 (2004), 205-220. doi: 10.1016/j.jcp.2004.02.005.
[16]	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.
[17]	D. Natroshvili, S. Kharibegashvili and Z. Tediashvili, Direct and inverse fluid-structure interaction problems, Rendiconti di Matematica, Serie VII, 20 (2000), 57-92.

show all references

References:

[1]	F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects, Inv. Prob., 22 (2006), 845-867.
[2]	F. Cakoni and H. Haddar, "A New Linear Sampling Method for the Electromagnetic Imagining of Buried Objects," in Mathematical methods in scattering theory and biomedical engineering, World Sci. Publ., Hackensack, NJ, 2006, 19-30. doi: 10.1142/9789812773197_0003.
[3]	D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Rev., 42 (2000), 369-414. doi: 10.1137/S0036144500367337.
[4]	D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inv. Prob., 21 (2005), 383-398.
[5]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2^nd edition, Springer-Verlag, New York, 1998.
[6]	D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inv. Prob., 13 (1997), 1477-1493.
[7]	J. Elschner, G. Hsiao and A. Rathsfeld, An inverse problem for fluid-solid interaction, Inverse Problems and Imaging, 2 (2007), 83-119.
[8]	J. Elschner, G. Hsiao and A. Rathsfeld, An optimization method in inverse acoustic scattering by an elastic obstacle, SIAM J. Appl. Math., 70 (2009), 168-187. doi: 10.1137/080736922.
[9]	J. Elschner, G. Hsiao and A. Rathsfeld, Comparison of numerical methods for the reconstruction of elastic obstacles from the far-field data of scattered acoustic waves, WAIS preprint No. 1479, 2010
[10]	T. Hargé, Valeurs propres d'un corps élastique, C. R. Acad. Sci. Paris, Sér. I Math., 311 (1990), 857-859.
[11]	G. Hsiao, R. Kleinman and G. F.Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., (2000), 139-163.
[12]	T. Huttunen, J. Kaipio and P. Monk, An ultra-weak method for acoustic fluid-solid interaction, J. Comput. Appl. Math., 213 (2008), 166-185. doi: 10.1016/j.cam.2006.12.030.
[13]	A. Kirsch and R. Kress, An optimization method in inverse acoustic scattering, in Boundary Elements IX (eds. C. Brebbia, W. Wendland and G. Kuhn), Springer, Heidelberg, 1987, 3-18.
[14]	C. 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.
[15]	A. Márquez, S. Meddahi and V. Selgas, A new BEM-FEM coupling strategy for two-dimensional fluid-solid interaction problems, J. Comput. Phys., 199 (2004), 205-220. doi: 10.1016/j.jcp.2004.02.005.
[16]	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.
[17]	D. Natroshvili, S. Kharibegashvili and Z. Tediashvili, Direct and inverse fluid-structure interaction problems, Rendiconti di Matematica, Serie VII, 20 (2000), 57-92.

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