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Some recent progress on inverse scattering problems within general polyhedral geometry

  • Received: 01 May 2020 Revised: 01 June 2020 Published: 24 August 2020
  • 35P05, 35P25, 35R30, 35Q60, 78A46

  • 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[J]. Electronic Research Archive, 2021, 29(1): 1753-1782. doi: 10.3934/era.2020090

    Related Papers:

  • 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.



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    [1] A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent. Proc. Roy. Soc. Edinburgh Sect. A (1978/79) 82: 251-272.
    [2] Determining a sound-soft polyhedral scatterer by a single far-field measurement. Proc. Amer. Math. Soc. (2005) 133: 1685-1691.
    [3] G. Alessandrini and L. Rondi, Corrigendum to "Determining a sound-soft polyhedral scatterer by a single far-field measurement", preprint, arXiv: math/0601406.
    [4] Uniqueness theorems for an inverse problem in a doubly periodic structure. Inverse Problems (1995) 11: 823-833.
    [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
    [6] Unique determination of periodic polyhedral structures by scattered electromagnetic fields. Trans. Amer. Math. Soc. (2011) 363: 4527-4551.
    [7] An inverse problem for scattering by a doubly periodic structure. Trans. Amer. Math. Soc. (1998) 350: 4089-4103.
    [8] Nonradiating sources and transmission eigenfunctions vanish at corners and edges. SIAM J. Math. Anal. (2018) 50: 6255-6270.
    [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
    [10] E. Blåsten and H. Liu, On corners scattering stably and stable shape determination by a single far-field pattern, preprint, arXiv: 1611.03647.
    [11] On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. (2017) 273: 3616-3632.
    [12] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815.
    [13] E. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, preprint, arXiv: 1808.01425.
    [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
    [15] Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-Posed Probl. (2008) 16: 19-33.
    [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
    [17] The determination of the surface impedance of a partially coated obstacle from far field data. SIAM J. Appl. Math. (2003/04) 64: 709-723.
    [18] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31230-7
    [19] The direct and inverse scattering problems for partially coated obstacles. Inverse Problems (2001) 17: 1997-2015.
    [20] The electromagnetic inverse-scattering problem for partly coated Lipschitz domains. Proc. Roy. Soc. Edinburgh Sect. A (2004) 134: 661-682.
    [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.
    [22] X. Cao, H. Diao and H. Liu, Determining a piecewise conductive medium body by a single far-field measurement, preprint, arXiv: 2005.04420.
    [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.
    [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.
    [25] Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves. Inverse Problems (2003) 19: 1361-1384.
    [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
    [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
    [28] Looking back on inverse scattering theory. SIAM Rev. (2018) 60: 779-807.
    [29] Uniqueness theorems for the inverse problem of acoustic scattering. IMA J. Appl. Math. (1983) 31: 253-259.
    [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.
    [31] Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number. J. Inverse Ill-Posed Probl. (2003) 11: 235-244.
    [32] An inverse problem in periodic diffractive optics: Global uniqueness with a single wavenumber. Inverse Problems (2003) 19: 779-787.
    [33] Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave. Inverse Problems (2006) 22: 355-364.
    [34] Schiffer's theorem in inverse scattering for periodic structures. Inverse Problems (1997) 13: 351-361.
    [35] Shape identification in inverse medium scattering problems with a single far-field pattern. SIAM J. Math. Anal. (2016) 48: 152-165.
    [36] An augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain. J. Math. Sci. (New York) (2002) 111: 3657-3666.
    [37] The denseness of the far field patterns for the transmission problem. IMA J. Appl. Math. (1986) 37: 213-225.
    [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
    [39] Uniqueness theorems in inverse scattering theory for periodic structures. Inverse Problems (1994) 10: 145-152.
    [40] Uniqueness in inverse obstacle scattering. Inverse Problems (1993) 9: 285-299.
    [41] P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, 26, Academic Press, New York-London, 1967.
    [42] Inverse obstacle problem: Local uniqueness for rougher obstacles and the identification of a ball. Inverse Problems (1997) 13: 1063-1069.
    [43] C. Liu and A. Nachman, A scattering theory analogue of a theorem of Polya and an inverse obstacle problem, in progress.
    [44] Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements. J. Differential Equations (2017) 262: 1631-1670.
    [45] 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) (2019) 21: 2945-2993.
    [46] Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers. Inverse Problems (2006) 22: 515-524.
    [47] On unique determination of partially coated polyhedral scatterers with far field measurements. Inverse Problems (2007) 23: 297-308.
    [48] (2000) Strongly Elliptic Systems and Boundary Integral Equation.Cambridge University Press.
    [49] On the Rayleigh assumption in scattering by a periodic surface. Proc. Cambridge Philos. Soc. (1969) 65: 773-791.
    [50] On the Rayleigh assumption in scattering by a periodic surface. II. Proc. Cambridge Philos. Soc. (1971) 69: 217-225.
    [51] Reconstructions from boundary measurements. Ann. of Math. (1988) 128: 531-576.
    [52] Multidimensional inverse spectral problems for the equation $-\Delta \psi+(v(x)-E u(x))\psi=0$. Func. Anal. Appl. (1988) 22: 263-272.
    [53] Recovery of the potential from fixed-energy scattering data. Inverse Problems (1988) 4: 877-886.
    [54] Stable determination of sound-soft polyhedral scatterers by a single measurement. Indiana Univ. Math. J. (2008) 57: 1377-1408.
    [55] The scattering of plane waves from periodic surfaces. Ann. Physics (1965) 33: 400-427.
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