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# Imaging junctions of waveguides

• * Corresponding author: Laurent Bourgeois
• 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.

Mathematics Subject Classification: Primary: 35J05, 35R25, 35R30, 35R35, 47G10.

 Citation: • • 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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