doi: 10.3934/ipi.2021069
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The interior inverse scattering problem for a two-layered cavity using the Bayesian method

1. 

School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun, 130022, China

2. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China

* Corresponding author: Weishi Yin

Received  June 2021 Revised  September 2021 Early access November 2021

In this paper, the Bayesian method is proposed for the interior inverse scattering problem to reconstruct the interface of a two-layered cavity. The scattered field is measured by the point sources located on a closed curve inside the interior interface. The well-posedness of the posterior distribution in the Bayesian framework is proved. The Markov Chain Monte Carlo algorithm is employed to explore the posterior density. Some numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Citation: 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, doi: 10.3934/ipi.2021069
References:
[1]

H. AmmariE. Iakovleva and D. Lesselier, A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency, Multiscale Model. Simul., 3 (2005), 597-628.  doi: 10.1137/040610854.

[2]

Z. Bai, H. Diao, H. Liu and Q. Meng, Effective medium theory for embedded obstacles in elasticity with applications to inverse problems, preprint, arXiv: 2102.09291.

[3]

T. Bui-Thanh and O. Ghattas, An analysis of infinite dimensional Bayesian inverse shape acoustic scattering and its numerical approximation, SIAM/ASA J. Uncertain. Quantif., 2 (2014), 203-222.  doi: 10.1137/120894877.

[4]

A. Carpio, S. Iakunin and G. Stadler, Bayesian approach to inverse scattering with topological priors, Inverse Problems, 36 (2020), 29pp. doi: 10.1088/1361-6420/abaa30.

[5]

Y. T. ChowY. DengY. HeH. Liu and X. Wang, Surface-localized transmission eigenstates, super-resolution imaging, and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), 946-975.  doi: 10.1137/20M1388498.

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^th$ edition, Applied Mathematical Sciences, 93, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[7]

Y. DengH. Liu and X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differential Equations, 267 (2019), 2192-2209.  doi: 10.1016/j.jde.2019.03.009.

[8]

J. HuangZ. Deng and L. Xu, Bayesian approach for inverse interior scattering problems with limited aperture, Appl. Anal., 98 (2020), 2802-2826.  doi: 10.1080/00036811.2020.1781828.

[9]

M. A. IglesiasY. Lu and A. Stuart, A Bayesian level set method for geometric inverse problems, Interfaces Free Bound., 18 (2016), 181-217.  doi: 10.4171/IFB/362.

[10]

S. Lasanen, Non-Gaussian statistical inverse problems. Part Ⅱ: Posterior convergence for approximated unknowns, Inverse Probl. Imaging., 6 (2012), 267-287.  doi: 10.3934/ipi.2012.6.267.

[11]

J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31 (2015), 28pp. doi: 10.1088/0266-5611/31/10/105006.

[12]

J. LiH. Liu and J. Zou, Multilevel linear sampling method for inverse scattering problems, SIAM J. Sci. Comput., 30 (2008), 1228-1250.  doi: 10.1137/060674247.

[13]

J. LiH. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009/10), 4013-4040.  doi: 10.1137/080734170.

[14]

Z. LiZ. Deng and J. Sun, Extended-sampling-Bayesian method for limited aperture inverse scattering problems, SIAM J. Imaging Sci., 13 (2020), 422-444.  doi: 10.1137/19M1270501.

[15]

Z. Li, Y. Liu, J. Sun and L. Xu, Quality-Bayesian approach to inverse acoustic source problems with partial data, SIAM J. Sci. Comput., 43 (2021), A1062–A1080. doi: 10.1137/20M132345X.

[16]

H. Liu and X. Liu, Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data, Inverse Problems, 33 (2017), 20pp. doi: 10.1088/1361-6420/aa6770.

[17]

H. LiuZ. ShangH. Sun and J. Zou, Singular perturbation of reduced wave equation and scattering from an embedded obstacle, J. Dynam. Differential Equations, 24 (2012), 803-821.  doi: 10.1007/s10884-012-9270-5.

[18]

H. LiuH. Zhao and C. Zou, Determining scattering support of anisotropic acoustic mediums and obstacles, Commun. Math. Sci., 13 (2015), 987-1000.  doi: 10.4310/CMS.2015.v13.n4.a7.

[19]

J. Liu, Y. Liu and J. Sun, An inverse medium problem using Stekloff eigenvalues and a Bayesian approach, Inverse Problems, 35 (2019), 20pp. doi: 10.1088/1361-6420/ab1be9.

[20]

Y. Liu, Y. Guo and J. Sun, A deterministic-statistical approach to reconstruct moving sources using sparse partial data, Inverse Problems, 37 (2021), 18pp. doi: 10.1088/1361-6420/abf813.

[21] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[22]

H.-H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 17pp. doi: 10.1088/0266-5611/27/3/035005.

[23]

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.

[24]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.

[25]

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.

[26]

Y. Wang, F. Ma and E. Zheng, Bayesian method for shape reconstruction in the inverse interior scattering problem, Math. Probl. Eng., 2015 (2015), 12pp. doi: 10.1155/2015/935294.

[27]

W. YinJ. GeP. Meng and F. Qu, A neural network method for the inverse scattering problem of impenetrable cavities, Electron. Res. Arch., 28 (2020), 1123-1142.  doi: 10.3934/era.2020062.

[28]

W. Yin, W. Yang and H. Liu, A neural network scheme for recovering scattering obstacles with limited phaseless far-field data, J. Comput. Phys., 417 (2020), 18pp. doi: 10.1016/j.jcp.2020.109594.

[29]

Y. Yin, W. Yin, P. Meng and H. Liu, On a hybrid approach for recovering multiple obstacles, in press, Commun. Comput. Phys., (2021).

[30]

F. ZengP. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Probl. Imaging, 7 (2013), 291-303.  doi: 10.3934/ipi.2013.7.291.

show all references

References:
[1]

H. AmmariE. Iakovleva and D. Lesselier, A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency, Multiscale Model. Simul., 3 (2005), 597-628.  doi: 10.1137/040610854.

[2]

Z. Bai, H. Diao, H. Liu and Q. Meng, Effective medium theory for embedded obstacles in elasticity with applications to inverse problems, preprint, arXiv: 2102.09291.

[3]

T. Bui-Thanh and O. Ghattas, An analysis of infinite dimensional Bayesian inverse shape acoustic scattering and its numerical approximation, SIAM/ASA J. Uncertain. Quantif., 2 (2014), 203-222.  doi: 10.1137/120894877.

[4]

A. Carpio, S. Iakunin and G. Stadler, Bayesian approach to inverse scattering with topological priors, Inverse Problems, 36 (2020), 29pp. doi: 10.1088/1361-6420/abaa30.

[5]

Y. T. ChowY. DengY. HeH. Liu and X. Wang, Surface-localized transmission eigenstates, super-resolution imaging, and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), 946-975.  doi: 10.1137/20M1388498.

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^th$ edition, Applied Mathematical Sciences, 93, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[7]

Y. DengH. Liu and X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differential Equations, 267 (2019), 2192-2209.  doi: 10.1016/j.jde.2019.03.009.

[8]

J. HuangZ. Deng and L. Xu, Bayesian approach for inverse interior scattering problems with limited aperture, Appl. Anal., 98 (2020), 2802-2826.  doi: 10.1080/00036811.2020.1781828.

[9]

M. A. IglesiasY. Lu and A. Stuart, A Bayesian level set method for geometric inverse problems, Interfaces Free Bound., 18 (2016), 181-217.  doi: 10.4171/IFB/362.

[10]

S. Lasanen, Non-Gaussian statistical inverse problems. Part Ⅱ: Posterior convergence for approximated unknowns, Inverse Probl. Imaging., 6 (2012), 267-287.  doi: 10.3934/ipi.2012.6.267.

[11]

J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31 (2015), 28pp. doi: 10.1088/0266-5611/31/10/105006.

[12]

J. LiH. Liu and J. Zou, Multilevel linear sampling method for inverse scattering problems, SIAM J. Sci. Comput., 30 (2008), 1228-1250.  doi: 10.1137/060674247.

[13]

J. LiH. Liu and J. Zou, Strengthened linear sampling method with a reference ball, SIAM J. Sci. Comput., 31 (2009/10), 4013-4040.  doi: 10.1137/080734170.

[14]

Z. LiZ. Deng and J. Sun, Extended-sampling-Bayesian method for limited aperture inverse scattering problems, SIAM J. Imaging Sci., 13 (2020), 422-444.  doi: 10.1137/19M1270501.

[15]

Z. Li, Y. Liu, J. Sun and L. Xu, Quality-Bayesian approach to inverse acoustic source problems with partial data, SIAM J. Sci. Comput., 43 (2021), A1062–A1080. doi: 10.1137/20M132345X.

[16]

H. Liu and X. Liu, Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data, Inverse Problems, 33 (2017), 20pp. doi: 10.1088/1361-6420/aa6770.

[17]

H. LiuZ. ShangH. Sun and J. Zou, Singular perturbation of reduced wave equation and scattering from an embedded obstacle, J. Dynam. Differential Equations, 24 (2012), 803-821.  doi: 10.1007/s10884-012-9270-5.

[18]

H. LiuH. Zhao and C. Zou, Determining scattering support of anisotropic acoustic mediums and obstacles, Commun. Math. Sci., 13 (2015), 987-1000.  doi: 10.4310/CMS.2015.v13.n4.a7.

[19]

J. Liu, Y. Liu and J. Sun, An inverse medium problem using Stekloff eigenvalues and a Bayesian approach, Inverse Problems, 35 (2019), 20pp. doi: 10.1088/1361-6420/ab1be9.

[20]

Y. Liu, Y. Guo and J. Sun, A deterministic-statistical approach to reconstruct moving sources using sparse partial data, Inverse Problems, 37 (2021), 18pp. doi: 10.1088/1361-6420/abf813.

[21] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[22]

H.-H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 17pp. doi: 10.1088/0266-5611/27/3/035005.

[23]

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.

[24]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.

[25]

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.

[26]

Y. Wang, F. Ma and E. Zheng, Bayesian method for shape reconstruction in the inverse interior scattering problem, Math. Probl. Eng., 2015 (2015), 12pp. doi: 10.1155/2015/935294.

[27]

W. YinJ. GeP. Meng and F. Qu, A neural network method for the inverse scattering problem of impenetrable cavities, Electron. Res. Arch., 28 (2020), 1123-1142.  doi: 10.3934/era.2020062.

[28]

W. Yin, W. Yang and H. Liu, A neural network scheme for recovering scattering obstacles with limited phaseless far-field data, J. Comput. Phys., 417 (2020), 18pp. doi: 10.1016/j.jcp.2020.109594.

[29]

Y. Yin, W. Yin, P. Meng and H. Liu, On a hybrid approach for recovering multiple obstacles, in press, Commun. Comput. Phys., (2021).

[30]

F. ZengP. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Probl. Imaging, 7 (2013), 291-303.  doi: 10.3934/ipi.2013.7.291.

Figure 1.  A schematic illustration of the two-layered cavity scattering problem
Figure 2.  Recoveries of the circle-shaped interface $ \Gamma_{0} $ by the Bayesian method with $ \sigma = 0.005,0.01,0.05 $, respectively
Figure 3.  The Markov chains of the coefficients $ q_{0},a_{1} $ and $ b_{1} $ with $ \sigma = 0.005,0.01,0.05 $, respectively
Figure 4.  Recoveries of the peanut-shaped interface $ \Gamma_{0} $ by the Bayesian method with $ \sigma = 0.005,0.01,0.05 $, respectively
Figure 5.  The Markov chains of the coefficients $ q_{0},a_{1} $ and $ b_{1} $ with $ \sigma = 0.005,0.01,0.05 $, respectively
Figure 6.  Recoveries of the peanut-shaped interface $ \Gamma_{0} $ by the Bayesian method with $ \rho = 0.3,0.2,0.1 $, respectively
Figure 7.  The Markov chains of the coefficients $ q_{0},a_{1} $ and $ b_{1} $ with $ \rho = 0.3,0.2,0.1 $, respectively
Figure 8.  Recoveries of the peanut-shaped interface $ \Gamma_{0} $ by the Bayesian method with $ \gamma_{1},\gamma_{2} $ and $ \gamma_{3} $, respectively
Figure 9.  The Markov chains of the coefficients $ q_{0},a_{1} $ and $ b_{1} $ with $ \gamma_{1},\gamma_{2} $ and $ \gamma_{3} $, respectively
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