American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inv. Prob., 21 (2005), 383-398.  Google Scholar

[5]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd edition, Springer-Verlag, New York, 1998.  Google Scholar

[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.  Google Scholar

[7]

J. Elschner, G. Hsiao and A. Rathsfeld, An inverse problem for fluid-solid interaction, Inverse Problems and Imaging, 2 (2007), 83-119.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[10]

T. Hargé, Valeurs propres d'un corps élastique, C. R. Acad. Sci. Paris, Sér. I Math., 311 (1990), 857-859.  Google Scholar

[11]

G. Hsiao, R. Kleinman and G. F.Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., (2000), 139-163.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inv. Prob., 21 (2005), 383-398.  Google Scholar

[5]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd edition, Springer-Verlag, New York, 1998.  Google Scholar

[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.  Google Scholar

[7]

J. Elschner, G. Hsiao and A. Rathsfeld, An inverse problem for fluid-solid interaction, Inverse Problems and Imaging, 2 (2007), 83-119.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[10]

T. Hargé, Valeurs propres d'un corps élastique, C. R. Acad. Sci. Paris, Sér. I Math., 311 (1990), 857-859.  Google Scholar

[11]

G. Hsiao, R. Kleinman and G. F.Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., (2000), 139-163.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

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