American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition
  • IPI Home
  • This Issue
  • Next Article
    Nonlocal TV-Gaussian prior for Bayesian inverse problems with applications to limited CT reconstruction
February  2020, 14(1): 133-152. doi: 10.3934/ipi.2019067

Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements

Houssem Haddar 1, and  Alexander Konschin 2,,

1. 

CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
2. 

Center for Industrial Mathematics, University of Bremen, Bibliothekstraẞe 5, 28359 Bremen, Germany

* Corresponding author: Alexander Konschin

Received  April 2019 Revised  August 2019 Published  November 2019

Fund Project: A. Konschin was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 281474342/GRK2224/1

We analyze the Factorization method to reconstruct the geometry of a local defect in a periodic absorbing layer using almost only incident plane waves at a fixed frequency. A crucial part of our analysis relies on the consideration of the range of a carefully designed far field operator, which characterizes the geometry of the defect. We further provide some validating numerical results in a two dimensional setting.

Keywords: Factorization method, inhomogeneous periodic media, far field measurements, reconstruction of defects, inverse problem.
Mathematics Subject Classification: Primary: 78A46, 35R30; Secondary: 35Q60.
Citation: Houssem Haddar, Alexander Konschin. Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements. Inverse Problems & Imaging, 2020, 14 (1) : 133-152. doi: 10.3934/ipi.2019067
References:
[1]

T. Arens and N. I. Grinberg, A complete factorization method for scattering by periodic structures, Computing, 75 (2005), 111-132.  doi: 10.1007/s00607-004-0092-0.  Google Scholar

[2]

T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures, Inverse Problems, 19 (2003), 1195-1211.  doi: 10.1088/0266-5611/19/5/311.  Google Scholar

[3]

L. Audibert, A. Girard and H. Haddar, Identifying defects in an unknown background using differential measurements, Inverse Probl. Imaging, 9 (2015), 625-643.  doi: 10.3934/ipi.2015.9.625.  Google Scholar

[4]

L. Audibert and H. Haddar, A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements, Inverse Problems, 30 (2014), 20pp.  doi: 10.1088/0266-5611/30/3/035011.  Google Scholar

[5]

A.-S. Bonnet-Ben Dhia, S. Fliss, C. Hazard and A. Tonnoir, A Rellich type theorem for the Helmholtz equation in a conical domain, C. R. Math. Acad. Sci. Paris, 354 (2016), 27-32.  doi: 10.1016/j.crma.2015.10.015.  Google Scholar

[6]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging, 7 (2013), 1123-1138.  doi: 10.3934/ipi.2013.7.1123.  Google Scholar

[7]

L. Bourgeois and S. Fliss, On the identification of defects in a periodic waveguide from far field data, Inverse Problems, 30 (2014), 31pp.  doi: 10.1088/0266-5611/30/9/095004.  Google Scholar

[8]

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

[9]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[10]

F. Cakoni, H. Haddar and T.-P. Nguyen, New interior transmission problem applied to a single Floquet-Bloch mode imaging of local perturbations in periodic media, Inverse Problems, 35 (2019), 31pp.  doi: 10.1088/1361-6420/aaecfd.  Google Scholar

[11]

D. L. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[12]

J. Elschner and G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions, Methods Appl. Anal., 18 (2011), 215-244.  doi: 10.4310/MAA.2011.v18.n2.a6.  Google Scholar

[13]

H. Haddar and T.-P. Nguyen, Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers, Comput. Math. Appl., 74 (2017), 2831-2855.  doi: 10.1016/j.camwa.2017.07.015.  Google Scholar

[14]

G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29 (2013), 25pp.  doi: 10.1088/0266-5611/29/11/115005.  Google Scholar

[15]

A. Kirsch and N. I. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications 36, Oxford University Press, Oxford, 2008. doi: 10.1093/acprof:oso/9780199213535.001.0001.  Google Scholar

[16]

A. Konschin and A. Lechleiter, Reconstruction of a local perturbation in inhomogeneous periodic layers from partial near field measurements, Inverse Problems, 35 (2019), 114006.  doi: 10.1088/1361-6420/ab1c66.  Google Scholar

[17]

A. Lechleiter, Imaging of periodic dielectrics, BIT, 50 (2010), 59-83.  doi: 10.1007/s10543-010-0255-7.  Google Scholar

[18]

A. Lechleiter, The Floquet-Bloch transform and scattering from locally perturbed periodic surfaces, J. Math. Anal. Appl., 446 (2017), 605-627.  doi: 10.1016/j.jmaa.2016.08.055.  Google Scholar

[19]

A. Lechleiter and D.-L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures, SIAM J. Imaging Sci., 6 (2013), 1111-1139.  doi: 10.1137/120903968.  Google Scholar

[20]

D.-L. Nguyen, Spectral methods for direct and inverse scattering from periodic structures, Ph.D thesis, Ecole Polytechnique, Palaiseau, France, 2012. Google Scholar

[21]

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media, Ph.D thesis, Karlsruher Institut für Technologie, 2010. doi: 10.5445/KSP/1000019400.  Google Scholar

[22]

J. Sarannen and G. Vainikko, Periodic Integral and Pseudodifferential Equations, Springer, Berlin, 2002. Google Scholar

[23]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, in Recent Advances in Scientific Computing and Applications, Contemp. Math., 586, Amer. Math. Soc., Providence, RI, 2013, 341–351. doi: 10.1090/conm/586/11652.  Google Scholar

[24]

J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 17pp.  doi: 10.1088/0266-5611/28/3/035004.  Google Scholar

show all references

References:
[1]

T. Arens and N. I. Grinberg, A complete factorization method for scattering by periodic structures, Computing, 75 (2005), 111-132.  doi: 10.1007/s00607-004-0092-0.  Google Scholar

[2]

T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures, Inverse Problems, 19 (2003), 1195-1211.  doi: 10.1088/0266-5611/19/5/311.  Google Scholar

[3]

L. Audibert, A. Girard and H. Haddar, Identifying defects in an unknown background using differential measurements, Inverse Probl. Imaging, 9 (2015), 625-643.  doi: 10.3934/ipi.2015.9.625.  Google Scholar

[4]

L. Audibert and H. Haddar, A generalized formulation of the linear sampling method with exact characterization of targets in terms of farfield measurements, Inverse Problems, 30 (2014), 20pp.  doi: 10.1088/0266-5611/30/3/035011.  Google Scholar

[5]

A.-S. Bonnet-Ben Dhia, S. Fliss, C. Hazard and A. Tonnoir, A Rellich type theorem for the Helmholtz equation in a conical domain, C. R. Math. Acad. Sci. Paris, 354 (2016), 27-32.  doi: 10.1016/j.crma.2015.10.015.  Google Scholar

[6]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging, 7 (2013), 1123-1138.  doi: 10.3934/ipi.2013.7.1123.  Google Scholar

[7]

L. Bourgeois and S. Fliss, On the identification of defects in a periodic waveguide from far field data, Inverse Problems, 30 (2014), 31pp.  doi: 10.1088/0266-5611/30/9/095004.  Google Scholar

[8]

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

[9]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[10]

F. Cakoni, H. Haddar and T.-P. Nguyen, New interior transmission problem applied to a single Floquet-Bloch mode imaging of local perturbations in periodic media, Inverse Problems, 35 (2019), 31pp.  doi: 10.1088/1361-6420/aaecfd.  Google Scholar

[11]

D. L. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[12]

J. Elschner and G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions, Methods Appl. Anal., 18 (2011), 215-244.  doi: 10.4310/MAA.2011.v18.n2.a6.  Google Scholar

[13]

H. Haddar and T.-P. Nguyen, Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers, Comput. Math. Appl., 74 (2017), 2831-2855.  doi: 10.1016/j.camwa.2017.07.015.  Google Scholar

[14]

G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29 (2013), 25pp.  doi: 10.1088/0266-5611/29/11/115005.  Google Scholar

[15]

A. Kirsch and N. I. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications 36, Oxford University Press, Oxford, 2008. doi: 10.1093/acprof:oso/9780199213535.001.0001.  Google Scholar

[16]

A. Konschin and A. Lechleiter, Reconstruction of a local perturbation in inhomogeneous periodic layers from partial near field measurements, Inverse Problems, 35 (2019), 114006.  doi: 10.1088/1361-6420/ab1c66.  Google Scholar

[17]

A. Lechleiter, Imaging of periodic dielectrics, BIT, 50 (2010), 59-83.  doi: 10.1007/s10543-010-0255-7.  Google Scholar

[18]

A. Lechleiter, The Floquet-Bloch transform and scattering from locally perturbed periodic surfaces, J. Math. Anal. Appl., 446 (2017), 605-627.  doi: 10.1016/j.jmaa.2016.08.055.  Google Scholar

[19]

A. Lechleiter and D.-L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures, SIAM J. Imaging Sci., 6 (2013), 1111-1139.  doi: 10.1137/120903968.  Google Scholar

[20]

D.-L. Nguyen, Spectral methods for direct and inverse scattering from periodic structures, Ph.D thesis, Ecole Polytechnique, Palaiseau, France, 2012. Google Scholar

[21]

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media, Ph.D thesis, Karlsruher Institut für Technologie, 2010. doi: 10.5445/KSP/1000019400.  Google Scholar

[22]

J. Sarannen and G. Vainikko, Periodic Integral and Pseudodifferential Equations, Springer, Berlin, 2002. Google Scholar

[23]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, in Recent Advances in Scientific Computing and Applications, Contemp. Math., 586, Amer. Math. Soc., Providence, RI, 2013, 341–351. doi: 10.1090/conm/586/11652.  Google Scholar

[24]

J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 17pp.  doi: 10.1088/0266-5611/28/3/035004.  Google Scholar

Figure 1.  Example for the refractive index $ n = n_p + q $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Three periods of the parameters and results for Example $ 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Three periods of main parameters for the numerical experiments
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Results for Example $ 2 $ with $ k^2 = 0.4 $, $ q = q_1 $, $ \nu = 10^{-3} $ and $ \nu = 10^{-6} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Results for Example $ 3 $ with $ k^2 = 1.8 $, $ \nu = 1 $ and $ \nu = 10^{-3} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Results for Example $ 4 $ with $ k^2 = 0.09 $, $ \nu = 10^{-3} $ and with $ k^2 = 3 $, $ \nu = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Results for FM and LSM with $ k^2 = 3 $ and $ 1\% $ relative noise
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

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
[2]

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
[3]

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
[4]

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
[5]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[6]

Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems & Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609
[7]

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
[8]

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
[9]

Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033
[10]

Lorenzo Audibert, Alexandre Girard, Houssem Haddar. Identifying defects in an unknown background using differential measurements. Inverse Problems & Imaging, 2015, 9 (3) : 625-643. doi: 10.3934/ipi.2015.9.625
[11]

Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems & Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048
[12]

Dinh-Liem Nguyen. The factorization method for the Drude-Born-Fedorov model for periodic chiral structures. Inverse Problems & Imaging, 2016, 10 (2) : 519-547. doi: 10.3934/ipi.2016010
[13]

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
[14]

Lauri Oksanen. Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements. Inverse Problems & Imaging, 2011, 5 (3) : 731-744. doi: 10.3934/ipi.2011.5.731
[15]

D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843
[16]

Hiroko Morimoto. Survey on time periodic problem for fluid flow under inhomogeneous boundary condition. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 631-639. doi: 10.3934/dcdss.2012.5.631
[17]

Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari. A note on analyticity properties of far field patterns. Inverse Problems & Imaging, 2013, 7 (2) : 491-498. doi: 10.3934/ipi.2013.7.491
[18]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205
[19]

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
[20]

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

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (31)
  • HTML views (34)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences