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doi: 10.3934/ipi.2022023
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## Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries

 1 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergstr. 69 A–4040 Linz, Austria 2 School of Mathematics, Jilin University, Changchun, Jilin 130012, China 3 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China 4 Department of Mathematics China, The Chinese University of Hong Kong, Shatin, Hong Kong, China

*Corresponding author: Hongyu Liu

Dedicated to the memory of Professor Victor Isakov (1947–2021)

Received  November 2021 Revised  March 2022 Early access April 2022

Fund Project: The first author was supported by the Austrian Science Fund (FWF): P 32660. The second author was supported in part by NSFC/RGC Joint Research Grant No. 12161160314 and the startup fund from Jilin University. The third author was supported by Hong Kong RGC General Research Funds (project numbers, 11300821, 12301218 and 12302919) and the NSFC/RGC Joint Research Grant (project number, N_CityU101/21). The fourth author was supported by Hong Kong RGC General Research Fund (projects 14306718 and 14306719)

We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in $\mathbb{R}^3$ by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results by a single measurement for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [12,13]. In this paper, we derive novel geometric properties that generalize and extend the related results in [13], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.

Citation: Xinlin Cao, Huaian Diao, Hongyu Liu, Jun Zou. Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries. Inverse Problems and Imaging, doi: 10.3934/ipi.2022023
##### 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. [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. [3] 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. [4] G. B. Arfken, H. J. Weber and F. E. Harris, Mathematical Methods for Physicists (Seventh Edition), Academic Press, New York-London 1966. doi: 10.1016/B978-0-12-384654-9.00014-1. [5] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 085005, 16 pp. doi: 10.1088/1361-6420/ab958f. [6] E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems, SIAM J. Math. Anal., 53 (2021), 3801-3837.  doi: 10.1137/20M1384002. [7] E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Anal. PDE, 14 (2021), 2207-2224.  doi: 10.2140/apde.2021.14.2207. [8] W. Bosch, On the computation of derivatives of Legendre functions, Phys. Chem. Earth, 25 (2000), 655-659.  doi: 10.1016/S1464-1895(00)00101-0. [9] M Cadilhac, Some mathematical aspects of the grating theory, Electromagnetic Theory of Gratings, Springer, (1980), 53–62. [10] X. Cao, H. Diao and J. Li, Some recent progress on inverse scattering problems within general polyhedral geometry, Electron. Res. Arch., 29 (2021), 1753-1782.  doi: 10.3934/era.2020090. [11] X. Cao, H. Diao and H. Liu, Determining a piecewise conductive medium body by a single far-field measurement, CSIAM Trans. Appl. Math., 1 (2020), 740-765. [12] 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., 143 (2020), 116-161.  doi: 10.1016/j.matpur.2020.09.011. [13] 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, SIAM J. Math. Anal., 53 (2021), 1263-1294.  doi: 10.1137/19M1292989. [14] X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011. [15] 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. [16] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, volume 93 of Applied Mathematical Sciences, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8. [17] D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Rev., 60 (2018), 779-807.  doi: 10.1137/17M1144763. [18] H. Diao, X. Cao and H. Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations, 46 (2021), 630-679.  doi: 10.1080/03605302.2020.1857397. [19] H. Diao, X. Fei, H. Liu and K. Yang, Visibility, invisibility and unique recovery of inverse electromagnetic problems with conical singularities, arXiv: 2204.02835, 2022. [20] H. Diao, H. Liu and B. Sun, On a local geometric property of the generalized elastic transmission eigenfunctions and application, Inverse Problems, 37 (2021), Paper No. 105015, 36 pp. doi: 10.1088/1361-6420/ac23c2. [21] H. Diao, H. Liu and L. Wang, On generalized Holmgren's principle to the Lamé operator with applications to inverse elastic problems, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 179, 50 pp. doi: 10.1007/s00526-020-01830-5. [22] H. Diao, H. Liu and L. Wang, Further results on generalized Holmgren's principle to the Lamé operator and applications, J. Differential Equations, 309 (2022), 841-882.  doi: 10.1016/j.jde.2021.11.039. [23] H. Diao, H. Liu, L. Zhang and J. Zou, Unique continuation from a generalized impedance edge-corner for Maxwell's system and applications to inverse problems, Inverse Problems, 37 (2021), Paper No. 035004, 32 pp. doi: 10.1088/1361-6420/abdb42. [24] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal periodic structures, Z. Anal. Anwend., 26 (2007), 165-177.  doi: 10.4171/ZAA/1316. [25] J. Elschner and M. Yamamoto, Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave, Inverse Problems, 24 (2008), 035004, 7 pp. doi: 10.1088/0266-5611/24/3/035004. [26] 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. [27] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 2006. [28] 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. [29] J. Li and H. Liu, Recovering a polyhedral obstacle by a few backscattering measurements, J. Differential Equations, 259 (2015), 2101-2120.  doi: 10.1016/j.jde.2015.03.030. [30] J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035011, 20 pp. doi: 10.1088/1361-6420/aa5bf3. [31] J. Li, H. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746.  doi: 10.1137/130907690. [32] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X. [33] J. Li, H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309.  doi: 10.1137/130920356. [34] H. Liu, A global uniqueness for formally determined inverse electromagnetic obstacle scattering, Inverse Problems, 24 (2008), 035018, 13 pp. doi: 10.1088/0266-5611/24/3/035018. [35] H. Liu, On local and global structures of transmission eigenfunctions and beyond, J. Inverse Ill-Posed Probl., 30 (2022), 287-305.  doi: 10.1515/jiip-2020-0099. [36] 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. [37] 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. [38] H. Liu and C.-H. Tsou, Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement, Inverse Problems, 36 (2020), 085010, 23 pp. doi: 10.1088/1361-6420/ab9d6b. [39] H. Liu and J. Xiao, Decoupling elastic waves and its applications, J. Differential Equations, 263 (2017), 4442-4480.  doi: 10.1016/j.jde.2017.05.022. [40] H. Liu, M. Yamamoto and J. Zou, Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering, Inverse Problems, 23 (2007), 2357-2366.  doi: 10.1088/0266-5611/23/6/005. [41] 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. [42] 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. [43] H. Liu and J. Zou, On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems, J. Phys.: Conf. Ser., 124 (2008), 012006.  doi: 10.1088/1742-6596/124/1/012006. [44] H. Liu and J. Zou, Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering, IMA J. Appl. Math., 72 (2007), 817-831.  doi: 10.1093/imamat/hxm013. [45] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equation, Cambridge University Press, Cambridge, 2000. [46] J.-C. Nédélec, Acoustic and Electromagnetic Equations, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7. [47] 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), 109594, 18pp. doi: 10.1016/j.jcp.2020.109594. [48] L. Rondi, Unique determination of non-smooth sound-soft scatterers by finitely many far-field measurements, Indiana Univ. Math. J., 52 (2003), 1631-1662.  doi: 10.1512/iumj.2003.52.2394. [49] 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. [50] L. Rondi, E. Sincich and M. Sini, Stable determination of a rigid scatterer in elastodynamics, SIAM J. Math. Anal., 53 (2021), 2660-2689.  doi: 10.1137/20M1352867.

show all references

##### 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. [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. [3] 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. [4] G. B. Arfken, H. J. Weber and F. E. Harris, Mathematical Methods for Physicists (Seventh Edition), Academic Press, New York-London 1966. doi: 10.1016/B978-0-12-384654-9.00014-1. [5] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 085005, 16 pp. doi: 10.1088/1361-6420/ab958f. [6] E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems, SIAM J. Math. Anal., 53 (2021), 3801-3837.  doi: 10.1137/20M1384002. [7] E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Anal. PDE, 14 (2021), 2207-2224.  doi: 10.2140/apde.2021.14.2207. [8] W. Bosch, On the computation of derivatives of Legendre functions, Phys. Chem. Earth, 25 (2000), 655-659.  doi: 10.1016/S1464-1895(00)00101-0. [9] M Cadilhac, Some mathematical aspects of the grating theory, Electromagnetic Theory of Gratings, Springer, (1980), 53–62. [10] X. Cao, H. Diao and J. Li, Some recent progress on inverse scattering problems within general polyhedral geometry, Electron. Res. Arch., 29 (2021), 1753-1782.  doi: 10.3934/era.2020090. [11] X. Cao, H. Diao and H. Liu, Determining a piecewise conductive medium body by a single far-field measurement, CSIAM Trans. Appl. Math., 1 (2020), 740-765. [12] 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., 143 (2020), 116-161.  doi: 10.1016/j.matpur.2020.09.011. [13] 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, SIAM J. Math. Anal., 53 (2021), 1263-1294.  doi: 10.1137/19M1292989. [14] X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011. [15] 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. [16] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^{th}$ edition, volume 93 of Applied Mathematical Sciences, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8. [17] D. Colton and R. Kress, Looking back on inverse scattering theory, SIAM Rev., 60 (2018), 779-807.  doi: 10.1137/17M1144763. [18] H. Diao, X. Cao and H. Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations, 46 (2021), 630-679.  doi: 10.1080/03605302.2020.1857397. [19] H. Diao, X. Fei, H. Liu and K. Yang, Visibility, invisibility and unique recovery of inverse electromagnetic problems with conical singularities, arXiv: 2204.02835, 2022. [20] H. Diao, H. Liu and B. Sun, On a local geometric property of the generalized elastic transmission eigenfunctions and application, Inverse Problems, 37 (2021), Paper No. 105015, 36 pp. doi: 10.1088/1361-6420/ac23c2. [21] H. Diao, H. Liu and L. Wang, On generalized Holmgren's principle to the Lamé operator with applications to inverse elastic problems, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 179, 50 pp. doi: 10.1007/s00526-020-01830-5. [22] H. Diao, H. Liu and L. Wang, Further results on generalized Holmgren's principle to the Lamé operator and applications, J. Differential Equations, 309 (2022), 841-882.  doi: 10.1016/j.jde.2021.11.039. [23] H. Diao, H. Liu, L. Zhang and J. Zou, Unique continuation from a generalized impedance edge-corner for Maxwell's system and applications to inverse problems, Inverse Problems, 37 (2021), Paper No. 035004, 32 pp. doi: 10.1088/1361-6420/abdb42. [24] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal periodic structures, Z. Anal. Anwend., 26 (2007), 165-177.  doi: 10.4171/ZAA/1316. [25] J. Elschner and M. Yamamoto, Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave, Inverse Problems, 24 (2008), 035004, 7 pp. doi: 10.1088/0266-5611/24/3/035004. [26] 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. [27] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 2006. [28] 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. [29] J. Li and H. Liu, Recovering a polyhedral obstacle by a few backscattering measurements, J. Differential Equations, 259 (2015), 2101-2120.  doi: 10.1016/j.jde.2015.03.030. [30] J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035011, 20 pp. doi: 10.1088/1361-6420/aa5bf3. [31] J. Li, H. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746.  doi: 10.1137/130907690. [32] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X. [33] J. Li, H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309.  doi: 10.1137/130920356. [34] H. Liu, A global uniqueness for formally determined inverse electromagnetic obstacle scattering, Inverse Problems, 24 (2008), 035018, 13 pp. doi: 10.1088/0266-5611/24/3/035018. [35] H. Liu, On local and global structures of transmission eigenfunctions and beyond, J. Inverse Ill-Posed Probl., 30 (2022), 287-305.  doi: 10.1515/jiip-2020-0099. [36] 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. [37] 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. [38] H. Liu and C.-H. Tsou, Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement, Inverse Problems, 36 (2020), 085010, 23 pp. doi: 10.1088/1361-6420/ab9d6b. [39] H. Liu and J. Xiao, Decoupling elastic waves and its applications, J. Differential Equations, 263 (2017), 4442-4480.  doi: 10.1016/j.jde.2017.05.022. [40] H. Liu, M. Yamamoto and J. Zou, Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering, Inverse Problems, 23 (2007), 2357-2366.  doi: 10.1088/0266-5611/23/6/005. [41] 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. [42] 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. [43] H. Liu and J. Zou, On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems, J. Phys.: Conf. Ser., 124 (2008), 012006.  doi: 10.1088/1742-6596/124/1/012006. [44] H. Liu and J. Zou, Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering, IMA J. Appl. Math., 72 (2007), 817-831.  doi: 10.1093/imamat/hxm013. [45] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equation, Cambridge University Press, Cambridge, 2000. [46] J.-C. Nédélec, Acoustic and Electromagnetic Equations, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7. [47] 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), 109594, 18pp. doi: 10.1016/j.jcp.2020.109594. [48] L. Rondi, Unique determination of non-smooth sound-soft scatterers by finitely many far-field measurements, Indiana Univ. Math. J., 52 (2003), 1631-1662.  doi: 10.1512/iumj.2003.52.2394. [49] 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. [50] L. Rondi, E. Sincich and M. Sini, Stable determination of a rigid scatterer in elastodynamics, SIAM J. Math. Anal., 53 (2021), 2660-2689.  doi: 10.1137/20M1352867.
A schematic illustration for an edge corner with the dihedral angle $\phi_0$
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