# American Institute of Mathematical Sciences

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  April 2021 Early access  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 and Imaging, 2021, 15 (2) : 285-314. doi: 10.3934/ipi.2020065
##### References:
 [1] 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. [2] V. Baronian, L. 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. [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. [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. [5] L. Borcea, F. 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. [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. [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. [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. [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. [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. [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. [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. [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. [14] F. Cakoni and D. Colton, Qualitative Methods In Inverse Scattering Theory, Springer-Verlag, Berlin, 2006. [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. [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. [17] D. Colton, M. 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. [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. [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. [20] C. Tsogka, D. 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. [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. [22] C. Tsogka, D. A. Mitsoudis and S. Papadimitropoulos, Imaging extended reflectors in a terminating waveguide, SIAM J. Imaging Sci., 11 (2018), 1680-1716.  doi: 10.1137/17M1159051.

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##### References:
 [1] 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. [2] V. Baronian, L. 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. [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. [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. [5] L. Borcea, F. 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. [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. [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. [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. [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. [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. [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. [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. [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. [14] F. Cakoni and D. Colton, Qualitative Methods In Inverse Scattering Theory, Springer-Verlag, Berlin, 2006. [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. [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. [17] D. Colton, M. 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. [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. [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. [20] C. Tsogka, D. 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. [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. [22] C. Tsogka, D. A. Mitsoudis and S. Papadimitropoulos, Imaging extended reflectors in a terminating waveguide, SIAM J. Imaging Sci., 11 (2018), 1680-1716.  doi: 10.1137/17M1159051.
Waveguide with an abrupt change of properties
Obstacles within the waveguide
Exact data on the section $\Sigma^0$. Left: $R^0 = 1.1$. Right: $R^0 = 3$
A waveguide with a transition zone (the domain $B_R$ is hatched)
A junction of three half-waveguides (the domain $B$ is hatched)
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
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
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
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
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
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
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
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
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
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 $\Sigma^2$
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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