Figure 1.
Waveguide with an abrupt change of properties
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Inverse scattering and stability for the biharmonic operator
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 |
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.
Keywords:
Linear Sampling method,
inverse scattering,
waveguide,
junction,
Dirichlet-to-Neumann operator,
fundamental solution.
Mathematics Subject Classification:
Primary: 35J05, 35R25, 35R30, 35R35, 47G10.
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
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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. |
show all references
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. |
Figure 2.
Obstacles within the waveguide
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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