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 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. CakoniD. 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. LiuX. 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. HuF. 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. SunY. 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. ZengP. 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. ZengX. LiuJ. 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. ZengX. LiuJ. 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. CakoniD. 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. LiuX. 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. HuF. 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. SunY. 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. ZengP. 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. ZengX. LiuJ. 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. ZengX. LiuJ. 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 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
[1]

Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062

[2]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[3]

Isaac Harris, Dinh-Liem Nguyen, Thi-Phong Nguyen. Direct sampling methods for isotropic and anisotropic scatterers with point source measurements. Inverse Problems and Imaging, 2022, 16 (5) : 1137-1162. doi: 10.3934/ipi.2022015

[4]

Yunwen Yin, Weishi Yin, Pinchao Meng, Hongyu Liu. The interior inverse scattering problem for a two-layered cavity using the Bayesian method. Inverse Problems and Imaging, 2022, 16 (4) : 673-690. doi: 10.3934/ipi.2021069

[5]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems and Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[6]

Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems and Imaging, 2021, 15 (5) : 1247-1267. doi: 10.3934/ipi.2021036

[7]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems and Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027

[8]

Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, 2022, 16 (5) : 1299-1323. doi: 10.3934/ipi.2022025

[9]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems and Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211

[10]

Guanghui Hu, Andrea Mantile, Mourad Sini, Tao Yin. Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles. Inverse Problems and Imaging, 2020, 14 (6) : 1025-1056. doi: 10.3934/ipi.2020054

[11]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems and Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035

[12]

Amin Boumenir, Vu Kim Tuan, Nguyen Hoang. The recovery of a parabolic equation from measurements at a single point. Evolution Equations and Control Theory, 2018, 7 (2) : 197-216. doi: 10.3934/eect.2018010

[13]

Tielei Zhu, Jiaqing Yang. A non-iterative sampling method for inverse elastic wave scattering by rough surfaces. Inverse Problems and Imaging, 2022, 16 (4) : 997-1017. doi: 10.3934/ipi.2022009

[14]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems and Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

[15]

Guanqiu Ma, Guanghui Hu. Factorization method for inverse time-harmonic elastic scattering with a single plane wave. Discrete and Continuous Dynamical Systems - B, 2022, 27 (12) : 7469-7492. doi: 10.3934/dcdsb.2022050

[16]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 1-21. doi: 10.3934/dcdss.2021006

[17]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems and Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

[18]

Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029

[19]

Alexey Penenko. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements. Inverse Problems and Imaging, 2020, 14 (5) : 757-782. doi: 10.3934/ipi.2020035

[20]

De-Han Chen, Daijun Jiang, Irwin Yousept, Jun Zou. Addendum to: "Variational source conditions for inverse Robin and flux problems by partial measurements". Inverse Problems and Imaging, 2022, 16 (2) : 481-481. doi: 10.3934/ipi.2022003

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (254)
  • HTML views (153)
  • Cited by (0)

Other articles
by authors

[Back to Top]