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# Extended sampling method for interior inverse scattering problems

• We consider an interior inverse scattering problem of reconstructing the shape of a cavity. The measurements are the scattered fields on a curve inside the cavity due to only one point source. In this paper, we employ the extending sampling method to reconstruct the cavity based on limited data. Numerical examples are provided to show the effectiveness of the method.

Mathematics Subject Classification: Primary: 45Q05, 65N20, 74J20.

 Citation: • • Figure 1.  Explicative figure. Relative location of the reference cavities $B_z$ and $D$

Figure 2.  Explicative figure. The reference cavities $B_z$ marked with different sampling points $z$ along different (left side) or same (right side) polar angle direction

Figure 3.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 4.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 5.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 6.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 7.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 8.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 9.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 10.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 11.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 12.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 13.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 14.  The contour plots of the indicator functions $I_1(z)$ (left) and $I_2(z)$ (right). Exact boundary $\partial D$, measurement location $C$ and point source $x_0$ are shown in red

Figure 15.  The contour plots of the indicator functions $I_1(z)$. Exact boundaries $\partial D$, measurement locations $C$ and point sources $x_0$ are shown in red

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