# American Institute of Mathematical Sciences

August  2020, 14(4): 719-731. doi: 10.3934/ipi.2020033

## Extended sampling method for interior inverse scattering problems

 College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

* Corresponding author: fzeng@cqu.edu.cn

Received  September 2019 Revised  February 2020 Published  May 2020

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.

Citation: Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems & Imaging, 2020, 14 (4) : 719-731. doi: 10.3934/ipi.2020033
##### References:
 [1] F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar [2] F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, Contemp. Math., 615 (2014), 71-88.  doi: 10.1090/conm/615/12246.  Google Scholar [3] D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1983.  Google Scholar [4] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2$^{nd}$ edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar [5] J. Elschner and G. Hu, Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves, Inverse Problems, 26 (2010), 115002, 23 pp. doi: 10.1088/0266-5611/26/11/115002.  Google Scholar [6] J. Liu, X. Liu and J. Sun, Extended sampling method for inverse elastic scattering problems using one incident wave, SIAM J. Imaging Sci., 12 (2019), 874-892.  doi: 10.1137/19M1237788.  Google Scholar [7] S. Meng, H. Haddar and F. Cakoni, The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30 (2014), 045008, 20 pp. doi: 10.1088/0266-5611/30/4/045008.  Google Scholar [8] Y. Hu, F. Cakoni and J. Liu, The inverse scattering problem for a partially coated cavity with interior measurements, Appl. Anal., 93 (2013), 936-956.  doi: 10.1080/00036811.2013.801458.  Google Scholar [9] G. Hu and X. Liu, Unique determination of balls and polyhedral scatterers with a single point source wave, Inverse Problems, 30 (2014), 065010, 14 pp. doi: 10.1088/0266-5611/30/6/065010.  Google Scholar [10] P. Jakubik and R. Potthast, Testing the integrity of some cavity – the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914.  doi: 10.1016/j.apnum.2007.04.007.  Google Scholar [11] X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18 pp. doi: 10.1088/0266-5611/30/1/015006.  Google Scholar [12] J. Liu and J. Sun, Extended sampling method in inverse scattering, Inverse Problems, 34 (2018), 085007, 17 pp. doi: 10.1088/1361-6420/aaca90.  Google Scholar [13] P. Li and Y. Wang, Near-field imaging of interior cavities, Commun. Comput. Phys., 17 (2015), 542-563.  doi: 10.4208/cicp.010414.250914a.  Google Scholar [14] H.-H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17 pp. doi: 10.1088/0266-5611/27/3/035005.  Google Scholar [15] H.-H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699-708.  doi: 10.1016/j.apnum.2010.10.011.  Google Scholar [16] H.-H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Adv. Comput. Math., 36 (2012), 157-174.  doi: 10.1007/s10444-011-9179-2.  Google Scholar [17] H.-H. Qin and X. Liu, The interior inverse scattering problem for cavities with an artificial obstacle, Appl. Numer. Math., 88 (2015), 18-30.  doi: 10.1016/j.apnum.2014.10.002.  Google Scholar [18] Y. Sun, Y. Guo and F. Ma, The reciprocity gap functional method for the inverse scattering problem for cavities, Appl. Anal., 95 (2016), 1327-1346.  doi: 10.1080/00036811.2015.1064519.  Google Scholar [19] F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002, 17 pp. doi: 10.1088/0266-5611/27/12/125002.  Google Scholar [20] F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Probl. and Imaging, 7 (2013), 291-303.  doi: 10.3934/ipi.2013.7.291.  Google Scholar [21] F. Zeng, X. Liu, J. Sun and L. Xu, The reciprocity gap method for a cavity in an inhomogeneous medium, Inverse Probl. Imaging, 10 (2016), 855-868.  doi: 10.3934/ipi.2016024.  Google Scholar [22] F. Zeng, X. Liu, J. Sun and L. Xu, Reciprocity gap method for an interior inverse scattering problem, J. Inverse Ill-Posed Probl., 25 (2017), 57-68.  doi: 10.1515/jiip-2015-0064.  Google Scholar

show all references

##### References:
 [1] F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar [2] F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, Contemp. Math., 615 (2014), 71-88.  doi: 10.1090/conm/615/12246.  Google Scholar [3] D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1983.  Google Scholar [4] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2$^{nd}$ edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar [5] J. Elschner and G. Hu, Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves, Inverse Problems, 26 (2010), 115002, 23 pp. doi: 10.1088/0266-5611/26/11/115002.  Google Scholar [6] J. Liu, X. Liu and J. Sun, Extended sampling method for inverse elastic scattering problems using one incident wave, SIAM J. Imaging Sci., 12 (2019), 874-892.  doi: 10.1137/19M1237788.  Google Scholar [7] S. Meng, H. Haddar and F. Cakoni, The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30 (2014), 045008, 20 pp. doi: 10.1088/0266-5611/30/4/045008.  Google Scholar [8] Y. Hu, F. Cakoni and J. Liu, The inverse scattering problem for a partially coated cavity with interior measurements, Appl. Anal., 93 (2013), 936-956.  doi: 10.1080/00036811.2013.801458.  Google Scholar [9] G. Hu and X. Liu, Unique determination of balls and polyhedral scatterers with a single point source wave, Inverse Problems, 30 (2014), 065010, 14 pp. doi: 10.1088/0266-5611/30/6/065010.  Google Scholar [10] P. Jakubik and R. Potthast, Testing the integrity of some cavity – the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914.  doi: 10.1016/j.apnum.2007.04.007.  Google Scholar [11] X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18 pp. doi: 10.1088/0266-5611/30/1/015006.  Google Scholar [12] J. Liu and J. Sun, Extended sampling method in inverse scattering, Inverse Problems, 34 (2018), 085007, 17 pp. doi: 10.1088/1361-6420/aaca90.  Google Scholar [13] P. Li and Y. Wang, Near-field imaging of interior cavities, Commun. Comput. Phys., 17 (2015), 542-563.  doi: 10.4208/cicp.010414.250914a.  Google Scholar [14] H.-H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17 pp. doi: 10.1088/0266-5611/27/3/035005.  Google Scholar [15] H.-H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699-708.  doi: 10.1016/j.apnum.2010.10.011.  Google Scholar [16] H.-H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Adv. Comput. Math., 36 (2012), 157-174.  doi: 10.1007/s10444-011-9179-2.  Google Scholar [17] H.-H. Qin and X. Liu, The interior inverse scattering problem for cavities with an artificial obstacle, Appl. Numer. Math., 88 (2015), 18-30.  doi: 10.1016/j.apnum.2014.10.002.  Google Scholar [18] Y. Sun, Y. Guo and F. Ma, The reciprocity gap functional method for the inverse scattering problem for cavities, Appl. Anal., 95 (2016), 1327-1346.  doi: 10.1080/00036811.2015.1064519.  Google Scholar [19] F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002, 17 pp. doi: 10.1088/0266-5611/27/12/125002.  Google Scholar [20] F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Probl. and Imaging, 7 (2013), 291-303.  doi: 10.3934/ipi.2013.7.291.  Google Scholar [21] F. Zeng, X. Liu, J. Sun and L. Xu, The reciprocity gap method for a cavity in an inhomogeneous medium, Inverse Probl. Imaging, 10 (2016), 855-868.  doi: 10.3934/ipi.2016024.  Google Scholar [22] F. Zeng, X. Liu, J. Sun and L. Xu, Reciprocity gap method for an interior inverse scattering problem, J. Inverse Ill-Posed Probl., 25 (2017), 57-68.  doi: 10.1515/jiip-2015-0064.  Google Scholar
Explicative figure. Relative location of the reference cavities $B_z$ and $D$
Explicative figure. The reference cavities $B_z$ marked with different sampling points $z$ along different (left side) or same (right side) polar angle direction
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
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
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
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
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
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
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
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
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
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
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
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
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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