# 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 & Imaging, 2021, 15 (2) : 285-314. doi: 10.3934/ipi.2020065
##### References:

show all references

##### References:
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 $\Sigma^2$
 [1] Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems & Imaging, 2021, 15 (5) : 1247-1267. doi: 10.3934/ipi.2021036 [2] Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043 [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] Shixu Meng. A sampling type method in an electromagnetic waveguide. Inverse Problems & Imaging, 2021, 15 (4) : 745-762. doi: 10.3934/ipi.2021012 [5] Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001 [6] 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 [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] Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033 [9] Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745 [10] Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201 [11] Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009 [12] Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, 2021, 15 (5) : 1015-1033. doi: 10.3934/ipi.2021026 [13] Siamak RabieniaHaratbar. Inverse scattering and stability for the biharmonic operator. Inverse Problems & Imaging, 2021, 15 (2) : 271-283. doi: 10.3934/ipi.2020064 [14] Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139 [15] Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary. Inverse Problems & Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054 [16] Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959 [17] Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577 [18] 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 [19] 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 [20] Laurent Bourgeois, Arnaud Recoquillay. The Linear Sampling Method for Kirchhoff-Love infinite plates. Inverse Problems & Imaging, 2020, 14 (2) : 363-384. doi: 10.3934/ipi.2020016

2020 Impact Factor: 1.639