April  2021, 15(2): 285-314. doi: 10.3934/ipi.2020065

Imaging junctions of waveguides

1. 

Laboratoire POEMS, ENSTA Paris, 828 Boulevard des Maréchaux, 91120 Palaiseau, France

2. 

Université Paris-Saclay, CEA, LIST, F-91120 Palaiseau, France

* Corresponding author: Laurent Bourgeois

Received  April 2020 Revised  September 2020 Published  November 2020

In this paper we address the identification of defects by the Linear Sampling Method in half-waveguides which are related to each other by junctions. Firstly a waveguide which is characterized by an abrupt change of properties is considered, secondly the more difficult case of several half-waveguides related to each other by a junction of complex geometry. Our approach is illustrated by some two-dimensional numerical experiments.

Citation: Laurent Bourgeois, Jean-François Fritsch, Arnaud Recoquillay. Imaging junctions of waveguides. Inverse Problems & Imaging, 2021, 15 (2) : 285-314. doi: 10.3934/ipi.2020065
References:
[1]

L. AudibertA. 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

[2]

V. BaronianL. Bourgeois and A. Recoquillay, Imaging an acoustic waveguide from surface data in the time domain, Wave Motion, 66 (2016), 68-87.  doi: 10.1016/j.wavemoti.2016.05.006.  Google Scholar

[3]

V. Baronian, L. Bourgeois, B. Chapuis and A. Recoquillay, Linear sampling method applied to non destructive testing of an elastic waveguide: theory, numerics and experiments, Inverse Problems, 34 (2018), 075006, 34 pp. doi: 10.1088/1361-6420/aac21e.  Google Scholar

[4]

A.-S. Bonnet-Bendhia and A. Tillequin, A generalized mode matching method for scattering problems with unbounded obstacles, Journal of Computational Acoustics, 9 (2001), 1611-1631.  doi: 10.1142/S0218396X01001005.  Google Scholar

[5]

L. BorceaF. Cakoni and S. Meng, A direct approach to imaging in a waveguide with perturbed geometry, J. Comput. Phys., 392 (2019), 556-577.  doi: 10.1016/j.jcp.2019.04.072.  Google Scholar

[6]

L. Borcea and S. Meng, Factorization method versus migration imaging in a waveguide, Inverse Problems, 35 (2019), 0124006, 33 pp. doi: 10.1088/1361-6420/ab2c9b.  Google Scholar

[7]

L. Borcea and D.-L. Nguyen, Imaging with electromagnetic waves in terminating waveguides, Inverse Probl. Imaging, 10 (2016), 915-941.  doi: 10.3934/ipi.2016027.  Google Scholar

[8]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: a modal formulation, Inverse Problems, 24 (2008), 015018, 20 pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar

[9]

L. Bourgeois, F. Le Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides, Inverse Problems, 27 (2011), 055001, 27 pp. doi: 10.1088/0266-5611/27/5/055001.  Google Scholar

[10]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 025017, 19 pp. doi: 10.1088/0266-5611/29/2/025017.  Google Scholar

[11]

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

[12]

L. Bourgeois and E. Lunéville, On the use of sampling methods to identify cracks in acoustic waveguides, Inverse Problems, 28 (2012), 105011, 18 pp. doi: 10.1088/0266-5611/28/10/105011.  Google Scholar

[13]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A formulation based on modes, Journal of Physics: Conference Series, 135 (2008), 012023. doi: 10.1088/1742-6596/135/1/012023.  Google Scholar

[14]

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

[15]

A. Charalambopoulos, D. Gintides, K. Kiriaki and A. Kirsch, The factorization method for an acoustic wave guide, in Mathematical Methods in Scattering Theory and Biomedical Engineering, World Sci. Publ., Hackensack, NJ, (2006), 120–127. doi: 10.1142/9789812773197_0013.  Google Scholar

[16]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[17]

D. ColtonM. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1997), 1477-1493.  doi: 10.1088/0266-5611/13/6/005.  Google Scholar

[18]

P. Monk and V. Selgas, An inverse acoustic waveguide problem in the time domain, Inverse Problems, 32 (2016), 055001, 26 pp. doi: 10.1088/0266-5611/32/5/055001.  Google Scholar

[19]

P. Monk, V. Selgas and F. Yang, Near-field linear sampling method for an inverse problem in an electromagnetic waveguide, Inverse Problems, 35 (2019), 065001, 27 pp. doi: 10.1088/1361-6420/ab0cdc.  Google Scholar

[20]

C. TsogkaD. A. Mitsoudis and S. Papadimitropoulos, Selective imaging of extended reflectors in two-dimensional waveguides, SIAM J. Imaging Sci., 6 (2013), 2714-2739.  doi: 10.1137/130924238.  Google Scholar

[21]

C. Tsogka, D. A. Mitsoudis and S. Papadimitropoulos, Partial-aperture array imaging in acoustic waveguides, Inverse Problems, 32 (2016), 125011, 31pp. doi: 10.1088/0266-5611/32/12/125011.  Google Scholar

[22]

C. TsogkaD. A. Mitsoudis and S. Papadimitropoulos, Imaging extended reflectors in a terminating waveguide, SIAM J. Imaging Sci., 11 (2018), 1680-1716.  doi: 10.1137/17M1159051.  Google Scholar

show all references

References:
[1]

L. AudibertA. 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

[2]

V. BaronianL. Bourgeois and A. Recoquillay, Imaging an acoustic waveguide from surface data in the time domain, Wave Motion, 66 (2016), 68-87.  doi: 10.1016/j.wavemoti.2016.05.006.  Google Scholar

[3]

V. Baronian, L. Bourgeois, B. Chapuis and A. Recoquillay, Linear sampling method applied to non destructive testing of an elastic waveguide: theory, numerics and experiments, Inverse Problems, 34 (2018), 075006, 34 pp. doi: 10.1088/1361-6420/aac21e.  Google Scholar

[4]

A.-S. Bonnet-Bendhia and A. Tillequin, A generalized mode matching method for scattering problems with unbounded obstacles, Journal of Computational Acoustics, 9 (2001), 1611-1631.  doi: 10.1142/S0218396X01001005.  Google Scholar

[5]

L. BorceaF. Cakoni and S. Meng, A direct approach to imaging in a waveguide with perturbed geometry, J. Comput. Phys., 392 (2019), 556-577.  doi: 10.1016/j.jcp.2019.04.072.  Google Scholar

[6]

L. Borcea and S. Meng, Factorization method versus migration imaging in a waveguide, Inverse Problems, 35 (2019), 0124006, 33 pp. doi: 10.1088/1361-6420/ab2c9b.  Google Scholar

[7]

L. Borcea and D.-L. Nguyen, Imaging with electromagnetic waves in terminating waveguides, Inverse Probl. Imaging, 10 (2016), 915-941.  doi: 10.3934/ipi.2016027.  Google Scholar

[8]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: a modal formulation, Inverse Problems, 24 (2008), 015018, 20 pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar

[9]

L. Bourgeois, F. Le Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides, Inverse Problems, 27 (2011), 055001, 27 pp. doi: 10.1088/0266-5611/27/5/055001.  Google Scholar

[10]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 025017, 19 pp. doi: 10.1088/0266-5611/29/2/025017.  Google Scholar

[11]

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

[12]

L. Bourgeois and E. Lunéville, On the use of sampling methods to identify cracks in acoustic waveguides, Inverse Problems, 28 (2012), 105011, 18 pp. doi: 10.1088/0266-5611/28/10/105011.  Google Scholar

[13]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A formulation based on modes, Journal of Physics: Conference Series, 135 (2008), 012023. doi: 10.1088/1742-6596/135/1/012023.  Google Scholar

[14]

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

[15]

A. Charalambopoulos, D. Gintides, K. Kiriaki and A. Kirsch, The factorization method for an acoustic wave guide, in Mathematical Methods in Scattering Theory and Biomedical Engineering, World Sci. Publ., Hackensack, NJ, (2006), 120–127. doi: 10.1142/9789812773197_0013.  Google Scholar

[16]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[17]

D. ColtonM. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1997), 1477-1493.  doi: 10.1088/0266-5611/13/6/005.  Google Scholar

[18]

P. Monk and V. Selgas, An inverse acoustic waveguide problem in the time domain, Inverse Problems, 32 (2016), 055001, 26 pp. doi: 10.1088/0266-5611/32/5/055001.  Google Scholar

[19]

P. Monk, V. Selgas and F. Yang, Near-field linear sampling method for an inverse problem in an electromagnetic waveguide, Inverse Problems, 35 (2019), 065001, 27 pp. doi: 10.1088/1361-6420/ab0cdc.  Google Scholar

[20]

C. TsogkaD. A. Mitsoudis and S. Papadimitropoulos, Selective imaging of extended reflectors in two-dimensional waveguides, SIAM J. Imaging Sci., 6 (2013), 2714-2739.  doi: 10.1137/130924238.  Google Scholar

[21]

C. Tsogka, D. A. Mitsoudis and S. Papadimitropoulos, Partial-aperture array imaging in acoustic waveguides, Inverse Problems, 32 (2016), 125011, 31pp. doi: 10.1088/0266-5611/32/12/125011.  Google Scholar

[22]

C. TsogkaD. A. Mitsoudis and S. Papadimitropoulos, Imaging extended reflectors in a terminating waveguide, SIAM J. Imaging Sci., 11 (2018), 1680-1716.  doi: 10.1137/17M1159051.  Google Scholar

Figure 1.  Waveguide with an abrupt change of properties
Figure 2.  Obstacles within the waveguide
Figure 15.  Exact data on the section $ \Sigma^0 $. Left: $ R^0 = 1.1 $. Right: $ R^0 = 3 $
Figure 3.  A waveguide with a transition zone (the domain $ B_R $ is hatched)
Figure 4.  A junction of three half-waveguides (the domain $ B $ is hatched)
Figure 5.  Full-scattering, $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 60 $ ($ \tilde{P} = 20 $). Top left: obstacle 3 and exact data. Top right: obstacle 3 and noisy data. Bottom left: obstacle 4 and exact data. Bottom right: obstacle 4 and noisy data
Figure 6.  Back-scattering for obstacle 1. Top left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 20 $ ($ \tilde{P} = 7 $), exact data. Top right: $ \kappa = 40 $ and $ \tilde{\kappa} = 20 $, noisy data. Middle left: $ \kappa = \tilde{\kappa} = 40 $ ($ P = \tilde{P} = 13 $), exact data. Middle right: $ \kappa = \tilde{\kappa} = 40 $, noisy data. Bottom left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 60 $ ($ \tilde{P} = 20 $), exact data. Bottom right: $ \kappa = 40 $ and $ \tilde{\kappa} = 60 $, noisy data
Figure 7.  Back-scattering for obstacle 2. Top left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 20 $ ($ P = 7 $), exact data. Top right: $ \kappa = 40 $ and $ \tilde{\kappa} = 20 $, noisy data. Middle left: $ \kappa = \tilde{\kappa} = 40 $ ($ P = \tilde{P} = 13 $), exact data. Middle right: $ \kappa = \tilde{\kappa} = 40 $, noisy data. Bottom left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 60 $ ($ \tilde{P} = 20 $), exact data. Bottom right: $ \kappa = 40 $ and $ \tilde{\kappa} = 60 $, noisy data
Figure 8.  Back-scattering for obstacle 4. Top left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 20 $ ($ \tilde{P} = 7 $), exact data. Top right: $ \kappa = 40 $ and $ \tilde{\kappa} = 20 $, noisy data. Middle left: $ \kappa = \tilde{\kappa} = 40 $ ($ P = \tilde{P} = 13 $), exact data. Middle right: $ \kappa = \tilde{\kappa} = 40 $, noisy data. Bottom left: $ \kappa = 40 $ ($ P = 13 $) and $ \tilde{\kappa} = 60 $ ($ \tilde{P} = 20 $), exact data. Bottom right: $ \kappa = 40 $ and $ \tilde{\kappa} = 60 $, noisy data
Figure 9.  Full-scattering, obstacle 3, $ \kappa = \tilde{\kappa} = 40 $, $ h = 0.65 $ ($ P = 9 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 13 $). Left: exact data. Right: noisy data
Figure 10.  Back-scattering for obstacle 1, $ \kappa = \tilde{\kappa} = 30 $ and $ h>\tilde{h} $. Top left: $ h = 1 $ ($ P = 10 $) and $ \tilde{h} = 0.5 $ ($ \tilde{P} = 5 $), exact data. Top right: $ h = 1 $ and $ \tilde{h} = 0.5 $, noisy data. Bottom left: $ h = 1 $ ($ P = 10 $) and $ \tilde{h} = 0.75 $ ($ \tilde{P} = 8 $), exact data. Bottom right: $ h = 1 $ and $ \tilde{h} = 0.75 $, noisy data
Figure 11.  Back-scattering for obstacle 1, $ \kappa = \tilde{\kappa} = 30 $ and $ h<\tilde{h} $. Top left: $ h = 0.5 $ ($ P = 5 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 10 $), exact data. Top right: $ h = 0.5 $ and $ \tilde{h} = 1 $, noisy data. Bottom left: $ h = 0.75 $ ($ P = 8 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 10 $), exact data. Bottom right: $ h = 0.75 $ and $ \tilde{h} = 1 $, noisy data
Figure 12.  Back-scattering for obstacle 2, $ \kappa = \tilde{\kappa} = 30 $ and $ h>\tilde{h} $. Top left: $ h = 1 $ ($ P = 10 $) and $ \tilde{h} = 0.5 $ ($ \tilde{P} = 5 $), exact data. Top right: $ h = 1 $ and $ \tilde{h} = 0.5 $, noisy data. Bottom left: $ h = 1 $ ($ P = 10 $) and $ \tilde{h} = 0.75 $ ($ \tilde{P} = 8 $), exact data. Bottom right: $ h = 1 $ and $ \tilde{h} = 0.75 $, noisy data
Figure 13.  Back-scattering for obstacle 2, $ \kappa = \tilde{\kappa} = 30 $ and $ h<\tilde{h} $. Top left: $ h = 0.5 $ ($ P = 5 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 10 $), exact data. Top right: $ h = 0.5 $ and $ \tilde{h} = 1 $, noisy data. Bottom left: $ h = 0.75 $ ($ P = 8 $) and $ \tilde{h} = 1 $ ($ \tilde{P} = 10 $), exact data. Bottom right: $ h = 0.75 $ and $ \tilde{h} = 1 $, noisy data
Figure 14.  Data on a single half-waveguide. Top left: exact data on section $ \Sigma^0 $. Top right: noisy data on section $ \Sigma^0 $. Middle left: exact data on section $ \Sigma^1 $. Middle right: noisy data on section $ \Sigma^1 $. Bottom left: exact data on section $ \Sigma^2 $. Bottom right: noisy data on section $ \Sigma^2 $
Figure 16.  Top: data on two half-waveguides. Left: exact data on sections $ \Sigma^0 $ and $ \Sigma^1 $. Right: noisy data on sections $ \Sigma^0 $ and $ \Sigma^1 $. Bottom: data on three half-waveguides. Left: exact data on sections $ \Sigma^0 $, $ \Sigma^1 $ and $ \Sigma^2 $. Right: noisy data on sections $ \Sigma^0 $, $ \Sigma^1 $ and $ \Sigma^2 $
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