Abstract
In this work, we examine the inverse problem to reconstruct the inner boundary of a cylindrical doubly-connected infinitely long medium from measurements of the scattered electromagnetic wave in the far-field. We consider the integral representation of the solution to derive a non-linear system of equations for the unknown radial function. We propose an iterative scheme using linearization and regularization techniques.
References
Colton D. Qualitative methods in inverse scattering theory. In: Engl HW, Louis AK, Rundell W, Eds. Inverse problems in medical imaging and nondestructive testing. Vienna: Springer; 1997. https://doi.org/10.1007/978-3-7091-6521-8_4
Kress R. Linear integral equations. vol. 82, 3rd ed. New York, NY: Springer; 2014. https://doi.org/10.1007/978-1-4614-9593-2
Kress R. A collocation method for a hypersingular boundary integral equation via trigonometric differentiation. J Integral Equations Appl. 2014; 26: 197-213. https://doi.org/10.1216/JIE-2014-26-2-197
Yousif HA, Köhler S. Scattering by two penetrable cylinders at oblique incidence I The analytical solution. J Opt Soc Am A. 1988; 5(7): 1085-96. https://doi.org/10.1364/JOSAA.5.001085
Wait JR. Scattering of a plane wave from a circular dielectric cylinder at oblique incidence. Can J Phys. 1955; 33: 189-95. https://doi.org/10.1139/p55-024
Nakamura G, Wang H. The direct electromagnetic scattering problem from an imperfectly conducting cylinder at oblique incidence. J Math Anal Appl. 2013; 397: 142-55. https://doi.org/10.1016/j.jmaa.2012.07.049
Wang H, Nakamura G. The integral equation method for electromagnetic scattering problem at oblique incidence. Appl Num Math. 2012; 62: 860-73. https://doi.org/10.1016/j.apnum.2012.02.006
Gintides D, Mindrinos L. The direct scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder. J Integral Equations Appl. 2016; 28: 91-122. https://doi.org/10.1216/JIE-2016-28-1-91
Gintides D, Mindrinos L. The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence. Appl Analysis. 2019; 98: 781-98. https://doi.org/10.1080/00036811.2017.1402891
Mindrinos L. The electromagnetic scattering problem by a cylindrical doubly connected domain at oblique incidence: the direct problem. IMA J Appl Math. 2019; 84: 292-311. https://doi.org/10.1093/imamat/hxy059
Kress R, Rundell W. Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl. 2005; 21: 1207-23. https://doi.org/10.1088/0266-5611/21/4/002
Johansson T, Sleeman BD. Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern. IMA J Appl Math. 2007; 72: 7296-112. https://doi.org/10.1093/imamat/hxl026
Altundag A, Kress R. On a two-dimensional inverse scattering problem for a dielectric. Appl Analysis. 2012; 91: 757-71. https://doi.org/10.1080/00036811.2011.619981
Chapko R, Gintides D, Mindrinos L. The inverse scattering problem by an elastic inclusion. Adv Comput Math. 2017; 44: 1-24. https://doi.org/10.1007/s10444-017-9550-z
Lee K-M. Inverse scattering problem from an impedance crack via a composite method. Wave Motion. 2015; 56: 43–51. https://doi.org/10.1016/j.wavemoti.2015.02.002
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright (c) 2023 Leonidas Mindrinos