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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 |
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.
Keywords:
Inverse scattering problem,
interior measurements,
extended sampling method,
single point source,
reference cavities.
Mathematics Subject Classification:
Primary: 45Q05, 65N20, 74J20.
Citation:
Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems and 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. |
| [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. |
| [3] |
D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1983. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18 pp.
doi: 10.1088/0266-5611/30/1/015006. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [3] |
D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1983. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18 pp.
doi: 10.1088/0266-5611/30/1/015006. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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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