American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions
  • IPI Home
  • This Issue
  • Next Article
    Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials
November  2013, 7(4): 1123-1138. doi: 10.3934/ipi.2013.7.1123

The factorization method applied to cracks with impedance boundary conditions

Yosra Boukari 1, and  Houssem Haddar 2,

1. 

INRIA Saclay Ile de France/Ecole Polytechnique, CMAP, Route de Saclay, 91128 Palaiseau Cedex, France
2. 

INRIA Saclay Ile de France / CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex

Received  August 2012 Revised  June 2013 Published  November 2013

We use the Factorization method to retrieve the shape of cracks with impedance boundary conditions from farfields associated with incident plane waves at a fixed frequency. This work is an extension of the study initiated by Kirsch and Ritter [Inverse Problems, 16, pp. 89-105, 2000] where the case of sound soft cracks is considered. We address here the scalar problem and provide theoretical validation of the method when the impedance boundary conditions hold on both sides of the crack. We then deduce an inversion algorithm and present some validating numerical results in the case of simply and multiply connected cracks.
Keywords: the Factorization method., sampling methods for cracks, Inverse scattering problems, impedance boundary conditions.
Mathematics Subject Classification: 35R30, 35J05, 78A4.
Citation: Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems & Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123
References:
[1]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proceedings of the American Mathematical Society, 133 (2005), 1685-1691, (electronic). doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar

[2]

H. Ammari, J. Garnier, H. Kang, W. K. Park and K. Solna, Imaging schemes for perfectly conducting cracks, SIAM, J. Appl. Math., 71 (2011), 68-91. doi: 10.1137/100800130.  Google Scholar

[3]

A. Ben Abda, F. Delbary and H. Haddar, On the use of the reciprocity-gap functional in inverse scattering from planar cracks, Math. Models Methods Appl. Sci.,15 (2005), 1553-1574. doi: 10.1142/S0218202505000819.  Google Scholar

[4]

F. Ben Hassen, Y. Boukari and H. Haddar, Application of the linear sampling method to retrieve cracks with impedance boundary conditions, Inverse Problems in Science and Engineering, 2012. doi: 10.1080/17415977.2012.686997.  Google Scholar

[5]

M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem. 35 (2011), 223-235. doi: 10.1016/j.enganabound.2010.08.007.  Google Scholar

[6]

M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal. 35 (2001), 595-605. doi: 10.1051/m2an:2001128.  Google Scholar

[7]

K. Bryan and M. S. Vogelius, A review of selected works on crack identification, in Geometric Methods in Inverse Problems and PDE Control (Springer, New York), MA Vol. 137, Math. Appl., 2004, 25-46. doi: 10.1007/978-1-4684-9375-7_3.  Google Scholar

[8]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, 2006.  Google Scholar

[9]

F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295. doi: 10.1088/0266-5611/19/2/303.  Google Scholar

[10]

N. Zeev and F. Cakoni, The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 69 (2009), 1024-1042. doi: 10.1137/070711542.  Google Scholar

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second edition, Heidelberg: Springer, 1998.  Google Scholar

[12]

D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math., 200 (2007), 21-31. doi: 10.1016/j.cam.2005.12.004.  Google Scholar

[13]

T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern, IMA Journal of Applied Mathematics, 72 (2007), 96-112. doi: 10.1093/imamat/hxl026.  Google Scholar

[14]

O. Ivanyshyn and R. Kress, Inverse scattering for planar cracks via nonlinear integral equations, Math. Methods Appl. Sci., 31 (2008), 1221-1232. doi: 10.1002/mma.970.  Google Scholar

[15]

N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions, J. Inverse Ill-Posed Probl., 10 (2002), 171-185. doi: 10.1515/jiip.2002.10.2.171.  Google Scholar

[16]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture in Mathematics and Its Applications 36, (2008).  Google Scholar

[17]

A. Kirsch and S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Problems, 16 (2000), 89-105. doi: 10.1088/0266-5611/16/1/308.  Google Scholar

[18]

R. Kress and P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems, 21 (2005), 773-784. doi: 10.1088/0266-5611/21/2/020.  Google Scholar

[19]

J. J. Liu, P. A. Krutitskii and M. Sini, Numerical solution of the scattering problem for acoustic waves by a two-sided crack in 2-dimensional space, J. Comput. Math., 29 (2011), 141-166. doi: 10.4208/jams.012111.012811a.  Google Scholar

[20]

A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 123-138. doi: 10.3934/ipi.2009.3.123.  Google Scholar

[21]

J. Liu and M. Sini, Reconstruction of cracks of different types from far-field measurements, Math. Meth. Appl. Sci., 33 (2010), 950-973. doi: 10.1002/mma.1203.  Google Scholar

[22]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.  Google Scholar

[23]

J. C. Nédélec, Acoustic and Electromagnetic Equations, Applied Matimatical Sciences. Springer-Verlag, Berlin, 2001.  Google Scholar

show all references

References:
[1]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proceedings of the American Mathematical Society, 133 (2005), 1685-1691, (electronic). doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar

[2]

H. Ammari, J. Garnier, H. Kang, W. K. Park and K. Solna, Imaging schemes for perfectly conducting cracks, SIAM, J. Appl. Math., 71 (2011), 68-91. doi: 10.1137/100800130.  Google Scholar

[3]

A. Ben Abda, F. Delbary and H. Haddar, On the use of the reciprocity-gap functional in inverse scattering from planar cracks, Math. Models Methods Appl. Sci.,15 (2005), 1553-1574. doi: 10.1142/S0218202505000819.  Google Scholar

[4]

F. Ben Hassen, Y. Boukari and H. Haddar, Application of the linear sampling method to retrieve cracks with impedance boundary conditions, Inverse Problems in Science and Engineering, 2012. doi: 10.1080/17415977.2012.686997.  Google Scholar

[5]

M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem. 35 (2011), 223-235. doi: 10.1016/j.enganabound.2010.08.007.  Google Scholar

[6]

M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal. 35 (2001), 595-605. doi: 10.1051/m2an:2001128.  Google Scholar

[7]

K. Bryan and M. S. Vogelius, A review of selected works on crack identification, in Geometric Methods in Inverse Problems and PDE Control (Springer, New York), MA Vol. 137, Math. Appl., 2004, 25-46. doi: 10.1007/978-1-4684-9375-7_3.  Google Scholar

[8]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, 2006.  Google Scholar

[9]

F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295. doi: 10.1088/0266-5611/19/2/303.  Google Scholar

[10]

N. Zeev and F. Cakoni, The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 69 (2009), 1024-1042. doi: 10.1137/070711542.  Google Scholar

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second edition, Heidelberg: Springer, 1998.  Google Scholar

[12]

D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math., 200 (2007), 21-31. doi: 10.1016/j.cam.2005.12.004.  Google Scholar

[13]

T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern, IMA Journal of Applied Mathematics, 72 (2007), 96-112. doi: 10.1093/imamat/hxl026.  Google Scholar

[14]

O. Ivanyshyn and R. Kress, Inverse scattering for planar cracks via nonlinear integral equations, Math. Methods Appl. Sci., 31 (2008), 1221-1232. doi: 10.1002/mma.970.  Google Scholar

[15]

N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions, J. Inverse Ill-Posed Probl., 10 (2002), 171-185. doi: 10.1515/jiip.2002.10.2.171.  Google Scholar

[16]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture in Mathematics and Its Applications 36, (2008).  Google Scholar

[17]

A. Kirsch and S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Problems, 16 (2000), 89-105. doi: 10.1088/0266-5611/16/1/308.  Google Scholar

[18]

R. Kress and P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems, 21 (2005), 773-784. doi: 10.1088/0266-5611/21/2/020.  Google Scholar

[19]

J. J. Liu, P. A. Krutitskii and M. Sini, Numerical solution of the scattering problem for acoustic waves by a two-sided crack in 2-dimensional space, J. Comput. Math., 29 (2011), 141-166. doi: 10.4208/jams.012111.012811a.  Google Scholar

[20]

A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 123-138. doi: 10.3934/ipi.2009.3.123.  Google Scholar

[21]

J. Liu and M. Sini, Reconstruction of cracks of different types from far-field measurements, Math. Meth. Appl. Sci., 33 (2010), 950-973. doi: 10.1002/mma.1203.  Google Scholar

[22]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.  Google Scholar

[23]

J. C. Nédélec, Acoustic and Electromagnetic Equations, Applied Matimatical Sciences. Springer-Verlag, Berlin, 2001.  Google Scholar

[1]

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
[2]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19
[3]

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

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
[5]

Lorenzo Audibert. The Generalized Linear Sampling and factorization methods only depends on the sign of contrast on the boundary. Inverse Problems & Imaging, 2017, 11 (6) : 1107-1119. doi: 10.3934/ipi.2017051
[6]

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
[7]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757
[8]

Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021036
[9]

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
[10]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[11]

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
[12]

Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
[13]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
[14]

Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709
[15]

Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems & Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103
[16]

Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107
[17]

Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951
[18]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[19]

Nuutti Hyvönen, Harri Hakula, Sampsa Pursiainen. Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 299-317. doi: 10.3934/ipi.2007.1.299
[20]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences