# American Institute of Mathematical Sciences

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, 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] Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258 [2] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [3] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [4] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [5] Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375 [6] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [7] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [8] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [9] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [10] Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108 [11] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [12] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [13] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [14] Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119 [15] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 [16] Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072 [17] Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074 [18] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168 [19] Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 [20] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

2019 Impact Factor: 1.373