# American Institute of Mathematical Sciences

doi: 10.3934/era.2020090

## Some recent progress on inverse scattering problems within general polyhedral geometry

 1 Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, China 2 School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China 3 School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, Shandong, China

* Corresponding author: Huaian Diao

Received  May 2020 Revised  June 2020 Published  August 2020

Unique identifiability by finitely many far-field measurements in the inverse scattering theory is a highly challenging fundamental mathematical topic. In this paper, we survey some recent progress on the inverse obstacle scattering problems and the inverse medium scattering problems associated with time-harmonic waves within a certain polyhedral geometry, where one can establish the unique identifiability results by finitely many measurements. Some unique identifiability issues on the inverse diffraction grating problems are also considered. Furthermore, the geometrical structures of Laplacian and transmission eigenfunctions are reviewed, which have important applications in the unique determination for inverse obstacle and medium scattering problems with finitely many measurements. We discuss the mathematical techniques and methods developed in the literature. Finally, we raise some intriguing open problems for the future investigation.

Citation: Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, doi: 10.3934/era.2020090
##### References:
 [1] H.-D. Alber, A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent, Proc. Roy. Soc. Edinburgh Sect. A, 82 (1978/79), 251-272.  doi: 10.1017/S0308210500011239.  Google Scholar [2] G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691.  doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar [3] G. Alessandrini and L. Rondi, Corrigendum to "Determining a sound-soft polyhedral scatterer by a single far-field measurement", preprint, arXiv: math/0601406. Google Scholar [4] H. Ammari, Uniqueness theorems for an inverse problem in a doubly periodic structure, Inverse Problems, 11 (1995), 823-833.  doi: 10.1088/0266-5611/11/4/013.  Google Scholar [5] G. Bao, L. Cowsar and W. 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Wang, On vanishing and localizing of transmission eigenfunctions near singular points: A numerical study, Inverse Problems, 33 (2017), 24pp. doi: 10.1088/1361-6420/aa8826.  Google Scholar [10] E. Blåsten and H. Liu, On corners scattering stably and stable shape determination by a single far-field pattern, preprint, arXiv: 1611.03647. Google Scholar [11] E. Blåsten and H. Liu, On vanishing near corners of transmission eigenfunctions, J. Funct. Anal., 273 (2017), 3616-3632.  doi: 10.1016/j.jfa.2017.08.023.  Google Scholar [12] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815. Google Scholar [13] E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, preprint, arXiv: 1808.01425. Google Scholar [14] O. Bondarenko and X. Liu, The factorization method for inverse obstacle scattering with conductive boundary condition, Inverse Problems, 29 (2013), 25pp. doi: 10.1088/0266-5611/29/9/095021.  Google Scholar [15] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.  doi: 10.1515/jiip.2008.002.  Google Scholar [16] M. Cadilhac, Some mathematical aspects of the grating theory, in Electromagnetic Theory of Gratings, Topics in Current Physics, 22, Springer, Berlin, Heidelberg, 1980, 53–62. doi: 10.1007/978-3-642-81500-3_2.  Google Scholar [17] F. Cakoni and D. Colton, The determination of the surface impedance of a partially coated obstacle from far field data, SIAM J. Appl. Math., 64 (2003/04), 709-723.  doi: 10.1137/S0036139903424254.  Google Scholar [18] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31230-7.  Google Scholar [19] F. Cakoni, D. Colton and P. Monk, The direct and inverse scattering problems for partially coated obstacles, Inverse Problems, 17 (2001), 1997-2015.  doi: 10.1088/0266-5611/17/6/327.  Google Scholar [20] F. Cakoni, D. Colton and P. Monk, The electromagnetic inverse-scattering problem for partly coated Lipschitz domains, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 661-682.  doi: 10.1017/S0308210500003413.  Google Scholar [21] F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications: Inside Out. II, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013, 529-580.  Google Scholar [22] X. Cao, H. Diao and H. Liu, Determining a piecewise conductive medium body by a single far-field measurement, preprint, arXiv: 2005.04420. Google Scholar [23] X. Cao, H. Diao, H. Liu and J. Zou, On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems, J. Math. Pures Appl., accepted, 2020. Google Scholar [24] X. Cao, H. Diao, H. Liu and J. Zou, On novel geometric structures of Laplacian eigenfunctions in $\mathbb{R}^3$ and applications to inverse problems, preprint, arXiv: 1909.10174. Google Scholar [25] J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems, 19 (2003), 1361-1384.  doi: 10.1088/0266-5611/19/6/008.  Google Scholar [26] J. Cheng and M. Yamamoto, Corrigendum: Uniqueness in an inverse scattering problem with non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems, 21 (2005). doi: 10.1088/0266-5611/21/3/C01.  Google Scholar [27] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar [28] D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Rev., 60 (2018), 779-807.  doi: 10.1137/17M1144763.  Google Scholar [29] D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259.  doi: 10.1093/imamat/31.3.253.  Google Scholar [30] H. Diao, X. Cao and H. Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, preprint, arXiv: 1811.01663. Google Scholar [31] J. Elschner, G. Schmidt and M. Yamamoto, Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number, J. Inverse Ill-Posed Probl., 11 (2003), 235-244.  doi: 10.1515/156939403769237024.  Google Scholar [32] J. Elschner, G. Schmidt and M. Yamamoto, An inverse problem in periodic diffractive optics: Global uniqueness with a single wavenumber, Inverse Problems, 19 (2003), 779-787.  doi: 10.1088/0266-5611/19/3/318.  Google Scholar [33] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave, Inverse Problems, 22 (2006), 355-364.  doi: 10.1088/0266-5611/22/1/019.  Google Scholar [34] F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures, Inverse Problems, 13 (1997), 351-361.  doi: 10.1088/0266-5611/13/2/010.  Google Scholar [35] G. Hu, M. Salo and and E. V. Vesalainen, Shape identification in inverse medium scattering problems with a single far-field pattern, SIAM J. Math. Anal., 48 (2016), 152-165.  doi: 10.1137/15M1032958.  Google Scholar [36] I. V. Kamotski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$ and S. A. Nazarov, An augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain, J. Math. Sci. (New York), 111 (2002), 3657-3666.  doi: 10.1023/A:1016377707919.  Google Scholar [37] A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), 213-225.  doi: 10.1093/imamat/37.3.213.  Google Scholar [38] A. Kirsch, Diffraction by periodic structures, in Inverse Problems in Mathematical Physics, Lecture Notes in Phys., 422, Springer, Berlin, 1993, 87–102. doi: 10.1007/3-540-57195-7_11.  Google Scholar [39] A. Kirsch, Uniqueness theorems in inverse scattering theory for periodic structures, Inverse Problems, 10 (1994), 145-152.  doi: 10.1088/0266-5611/10/1/011.  Google Scholar [40] A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), 285-299.  doi: 10.1088/0266-5611/9/2/009.  Google Scholar [41] P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, 26, Academic Press, New York-London, 1967.  Google Scholar [42] C. Liu, Inverse obstacle problem: Local uniqueness for rougher obstacles and the identification of a ball, Inverse Problems, 13 (1997), 1063-1069.  doi: 10.1088/0266-5611/13/4/011.  Google Scholar [43] C. Liu and A. Nachman, A scattering theory analogue of a theorem of Polya and an inverse obstacle problem, in progress. Google Scholar [44] H. Liu, M. Petrini, L. Rondi and J. Xiao, Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements, J. Differential Equations, 262 (2017), 1631-1670.  doi: 10.1016/j.jde.2016.10.021.  Google Scholar [45] H. Liu, L. Rondi and J. Xiao, Mosco convergence for $H(curl)$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.  Google Scholar [46] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar [47] H. Liu and J. Zou, On unique determination of partially coated polyhedral scatterers with far field measurements, Inverse Problems, 23 (2007), 297-308.  doi: 10.1088/0266-5611/23/1/016.  Google Scholar [48] W. McLean, Strongly Elliptic Systems and Boundary Integral Equation, Cambridge University Press, Cambridge, 2000.   Google Scholar [49] R. F. Millar, On the Rayleigh assumption in scattering by a periodic surface, Proc. Cambridge Philos. Soc., 65 (1969), 773-791.  doi: 10.1017/S0305004100003613.  Google Scholar [50] R. F. Millar, On the Rayleigh assumption in scattering by a periodic surface. II, Proc. Cambridge Philos. Soc., 69 (1971), 217-225.  doi: 10.1017/S0305004100046570.  Google Scholar [51] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [52] R. G. Novikov, Multidimensional inverse spectral problems for the equation $-\Delta \psi+(v(x)-E u(x))\psi=0$, Func. Anal. Appl., 22 (1988), 263-272.  doi: 10.1007/BF01077418.  Google Scholar [53] A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886.  doi: 10.1088/0266-5611/4/3/020.  Google Scholar [54] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, Indiana Univ. Math. J., 57 (2008), 1377-1408.  doi: 10.1512/iumj.2008.57.3217.  Google Scholar [55] J. L. Uretsky, The scattering of plane waves from periodic surfaces, Ann. Physics, 33 (1965), 400-427.  doi: 10.1016/0003-4916(65)90269-1.  Google Scholar

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##### References:
 [1] H.-D. Alber, A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent, Proc. Roy. Soc. Edinburgh Sect. A, 82 (1978/79), 251-272.  doi: 10.1017/S0308210500011239.  Google Scholar [2] G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691.  doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar [3] G. Alessandrini and L. Rondi, Corrigendum to "Determining a sound-soft polyhedral scatterer by a single far-field measurement", preprint, arXiv: math/0601406. Google Scholar [4] H. Ammari, Uniqueness theorems for an inverse problem in a doubly periodic structure, Inverse Problems, 11 (1995), 823-833.  doi: 10.1088/0266-5611/11/4/013.  Google Scholar [5] G. Bao, L. Cowsar and W. Masters, Mathematical Modeling in Optical Science, Frontiers in Applied Mathematics, 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898717594.  Google Scholar [6] G. Bao, H. Zhang and J. Zou, Unique determination of periodic polyhedral structures by scattered electromagnetic fields, Trans. Amer. Math. Soc., 363 (2011), 4527-4551.  doi: 10.1090/S0002-9947-2011-05334-1.  Google Scholar [7] G. Bao and Z. Zhou, An inverse problem for scattering by a doubly periodic structure, Trans. Amer. Math. Soc., 350 (1998), 4089-4103.  doi: 10.1090/S0002-9947-98-02227-2.  Google Scholar [8] E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.  doi: 10.1137/18M1182048.  Google Scholar [9] E. Blåsten, X. Li, H. Liu and Y. Wang, On vanishing and localizing of transmission eigenfunctions near singular points: A numerical study, Inverse Problems, 33 (2017), 24pp. doi: 10.1088/1361-6420/aa8826.  Google Scholar [10] E. Blåsten and H. Liu, On corners scattering stably and stable shape determination by a single far-field pattern, preprint, arXiv: 1611.03647. Google Scholar [11] E. Blåsten and H. Liu, On vanishing near corners of transmission eigenfunctions, J. Funct. Anal., 273 (2017), 3616-3632.  doi: 10.1016/j.jfa.2017.08.023.  Google Scholar [12] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815. Google Scholar [13] E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, preprint, arXiv: 1808.01425. Google Scholar [14] O. Bondarenko and X. Liu, The factorization method for inverse obstacle scattering with conductive boundary condition, Inverse Problems, 29 (2013), 25pp. doi: 10.1088/0266-5611/29/9/095021.  Google Scholar [15] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.  doi: 10.1515/jiip.2008.002.  Google Scholar [16] M. Cadilhac, Some mathematical aspects of the grating theory, in Electromagnetic Theory of Gratings, Topics in Current Physics, 22, Springer, Berlin, Heidelberg, 1980, 53–62. doi: 10.1007/978-3-642-81500-3_2.  Google Scholar [17] F. Cakoni and D. Colton, The determination of the surface impedance of a partially coated obstacle from far field data, SIAM J. Appl. Math., 64 (2003/04), 709-723.  doi: 10.1137/S0036139903424254.  Google Scholar [18] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31230-7.  Google Scholar [19] F. Cakoni, D. Colton and P. Monk, The direct and inverse scattering problems for partially coated obstacles, Inverse Problems, 17 (2001), 1997-2015.  doi: 10.1088/0266-5611/17/6/327.  Google Scholar [20] F. Cakoni, D. Colton and P. Monk, The electromagnetic inverse-scattering problem for partly coated Lipschitz domains, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 661-682.  doi: 10.1017/S0308210500003413.  Google Scholar [21] F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications: Inside Out. II, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013, 529-580.  Google Scholar [22] X. Cao, H. Diao and H. Liu, Determining a piecewise conductive medium body by a single far-field measurement, preprint, arXiv: 2005.04420. Google Scholar [23] X. Cao, H. Diao, H. Liu and J. Zou, On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems, J. Math. Pures Appl., accepted, 2020. Google Scholar [24] X. Cao, H. Diao, H. Liu and J. Zou, On novel geometric structures of Laplacian eigenfunctions in $\mathbb{R}^3$ and applications to inverse problems, preprint, arXiv: 1909.10174. Google Scholar [25] J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems, 19 (2003), 1361-1384.  doi: 10.1088/0266-5611/19/6/008.  Google Scholar [26] J. Cheng and M. Yamamoto, Corrigendum: Uniqueness in an inverse scattering problem with non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems, 21 (2005). doi: 10.1088/0266-5611/21/3/C01.  Google Scholar [27] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar [28] D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Rev., 60 (2018), 779-807.  doi: 10.1137/17M1144763.  Google Scholar [29] D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259.  doi: 10.1093/imamat/31.3.253.  Google Scholar [30] H. Diao, X. Cao and H. Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, preprint, arXiv: 1811.01663. Google Scholar [31] J. Elschner, G. Schmidt and M. Yamamoto, Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number, J. Inverse Ill-Posed Probl., 11 (2003), 235-244.  doi: 10.1515/156939403769237024.  Google Scholar [32] J. Elschner, G. Schmidt and M. Yamamoto, An inverse problem in periodic diffractive optics: Global uniqueness with a single wavenumber, Inverse Problems, 19 (2003), 779-787.  doi: 10.1088/0266-5611/19/3/318.  Google Scholar [33] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave, Inverse Problems, 22 (2006), 355-364.  doi: 10.1088/0266-5611/22/1/019.  Google Scholar [34] F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures, Inverse Problems, 13 (1997), 351-361.  doi: 10.1088/0266-5611/13/2/010.  Google Scholar [35] G. Hu, M. Salo and and E. V. Vesalainen, Shape identification in inverse medium scattering problems with a single far-field pattern, SIAM J. Math. Anal., 48 (2016), 152-165.  doi: 10.1137/15M1032958.  Google Scholar [36] I. V. Kamotski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$ and S. A. Nazarov, An augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain, J. Math. Sci. (New York), 111 (2002), 3657-3666.  doi: 10.1023/A:1016377707919.  Google Scholar [37] A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), 213-225.  doi: 10.1093/imamat/37.3.213.  Google Scholar [38] A. Kirsch, Diffraction by periodic structures, in Inverse Problems in Mathematical Physics, Lecture Notes in Phys., 422, Springer, Berlin, 1993, 87–102. doi: 10.1007/3-540-57195-7_11.  Google Scholar [39] A. Kirsch, Uniqueness theorems in inverse scattering theory for periodic structures, Inverse Problems, 10 (1994), 145-152.  doi: 10.1088/0266-5611/10/1/011.  Google Scholar [40] A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), 285-299.  doi: 10.1088/0266-5611/9/2/009.  Google Scholar [41] P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, 26, Academic Press, New York-London, 1967.  Google Scholar [42] C. Liu, Inverse obstacle problem: Local uniqueness for rougher obstacles and the identification of a ball, Inverse Problems, 13 (1997), 1063-1069.  doi: 10.1088/0266-5611/13/4/011.  Google Scholar [43] C. Liu and A. Nachman, A scattering theory analogue of a theorem of Polya and an inverse obstacle problem, in progress. Google Scholar [44] H. Liu, M. Petrini, L. Rondi and J. Xiao, Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements, J. Differential Equations, 262 (2017), 1631-1670.  doi: 10.1016/j.jde.2016.10.021.  Google Scholar [45] H. Liu, L. Rondi and J. Xiao, Mosco convergence for $H(curl)$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.  Google Scholar [46] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar [47] H. Liu and J. Zou, On unique determination of partially coated polyhedral scatterers with far field measurements, Inverse Problems, 23 (2007), 297-308.  doi: 10.1088/0266-5611/23/1/016.  Google Scholar [48] W. McLean, Strongly Elliptic Systems and Boundary Integral Equation, Cambridge University Press, Cambridge, 2000.   Google Scholar [49] R. F. Millar, On the Rayleigh assumption in scattering by a periodic surface, Proc. Cambridge Philos. Soc., 65 (1969), 773-791.  doi: 10.1017/S0305004100003613.  Google Scholar [50] R. F. Millar, On the Rayleigh assumption in scattering by a periodic surface. II, Proc. Cambridge Philos. Soc., 69 (1971), 217-225.  doi: 10.1017/S0305004100046570.  Google Scholar [51] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [52] R. G. Novikov, Multidimensional inverse spectral problems for the equation $-\Delta \psi+(v(x)-E u(x))\psi=0$, Func. Anal. Appl., 22 (1988), 263-272.  doi: 10.1007/BF01077418.  Google Scholar [53] A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886.  doi: 10.1088/0266-5611/4/3/020.  Google Scholar [54] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, Indiana Univ. Math. J., 57 (2008), 1377-1408.  doi: 10.1512/iumj.2008.57.3217.  Google Scholar [55] J. L. Uretsky, The scattering of plane waves from periodic surfaces, Ann. Physics, 33 (1965), 400-427.  doi: 10.1016/0003-4916(65)90269-1.  Google Scholar
Schematic illustration of the two polygonal geometries in $\mathbb{R}^2$ for a conductive medium body
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