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An Optimization Method in Inverse Elastic Scattering for One-Dimensional Grating Profiles

  • Johannes Elschner (a1) and Guanghui Hu (a1)


Consider the inverse diffraction problem to determine a two-dimensional periodic structure from scattered elastic waves measured above the structure. We formulate the inverse problem as a least squares optimization problem, following the two-step algorithm by G. Bruckner and J. Elschner [Inverse Probl., 19 (2003), 315-329] for electromagnetic diffraction gratings. Such a method is based on the Kirsch-Kress optimization scheme and consists of two parts: a linear severely ill-posed problem and a nonlinear well-posed one. We apply this method to both smooth (C2) and piecewise linear gratings for the Dirichlet boundary value problem of the Navier equation. Numerical reconstructions from exact and noisy data illustrate the feasibility of the method.


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[1]Antonios, C., Drossos, G.and Kiriakie, K., On the uniqueness of the inverse elastic scattering problem for periodic structures, Inverse Probl., 17 (2001), 19231935.
[2]Arens, T., The scattering of plane elastic waves by a one-dimensional periodic surface, Math. Meth. Appl. Sci., 22 (1999), 5572.
[3]Arens, T., Uniqueness for elastic wave scattering by rough surfaces, SIAM J. Math. Anal., 33 (2001), 461471.
[4]Arens, T., Existence of solution in elastic wave scattering by unbounded rough surfaces, Math. Meth. Appl. Sci., 25 (2001), 507528.
[5]Arens, T. and Grinberg, N., A complete factorization method for scattering by periodic surface, Computing, 75 (2005), 111132.
[6]Arens, T. and Kirsch, A., The factorization method in inverse scattering from periodic structures, Inverse Probl., 19 (2003), 11951211.
[7]Atkinson, K. E., A discrete Galerkin method for first kind integral equations with a logarithmic kernel, J. Int. Equations Appl., 1 (1988), 343363.
[8]Bao, G., Cowsar, L. and Master, W.s, eds., Mathematical Modeling in Optical Science, Philadelphia, USA: SIAM, 2001.
[9]Bruckner, G. and Elschner, J., A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles, Inverse Probl., 19 (2003), 315329.
[10]Bruckner, G. and Elschner, J., The numerical solution of an inverse periodic transmission problem, Math. Meth. Appl. Sci., 28 (2005), 757778.
[11]Bruckner,, G.Elschner, J. and Yamamoto, M., An optimization method for profile reconstruction, in: Progress in Analysis, Proceed. 3rd ISAAC congress, (Singapore: World Scientific) (2003), 13911404.
[12]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Berlin: Springer, 1998.
[13]Elschner, J. and Hu, G., Variational approach to scattering of plane elastic waves by diffraction gratings, Math. Meth. Appl. Sci., 33 (2010), 19241941.
[14]Elschner, J. and Hu, G., Inverse scattering of elastic waves by periodic structures: uniqueness under the third or fourth kind boundary conditions, Methods Appl. Anal., 18 (2011), 215244.
[15]Elschner and, J.Hu, G., Scattering of plane elastic waves by three-dimensional diffraction gratings, Math. Models Methods Appl. Sci., 22 (2012), 1150019.
[16]Elschner, J. and Hu, G., Uniqueness in inverse scattering of elastic waves by threedimensional polyhedral diffraction gratings, Inver. Ill Posed Prob., 19 (2011), 717768.
[17]Elschner, J.and Stephan, E. P., A discrete collocation method for Symm's integral equation on curves with corners, J. Comput. Appl. Math., 75 (1996), 131146.
[18]Elschner, J. and Yamamoto, M., An inverse problem in periodic diffractive optics: reconstruction of Lipschitz grating profiles, Appl. Anal., 81 (2002), 13071328.
[19]Hettlich, F., Iterative regularization schemes in inverse scattering by periodic structures, Inverse Probl., 18 (2002), 701714.
[20]Hettlich, F. and Kirsch, A., Schiffer's theorem in inverse scattering for periodic structures, Inverse Probl., 13 (1997), 351361.
[21]Ito, K. and Reitich, F., A high-order perturbation approach to profile reconstruction I: perfectly conducting grating, Inverse Probl., 15 (1999), 10671085.
[22]Kress, R., Inverse elastic scattering from a crack, Inverse Probl., 12 (1996), 667684.
[23]Linton, C. M., The Green's function for the two-dimensional Helmholtz equation in periodic domains, J. Eng. Math., 33 (1998), 377401.
[24]Rathsfeld, A., Schmidt, G. and Kleemann, B. H., On a fast integral equation method for diffraction gratings, Commun. Comput. Phys., 1 (2006), 9841009.
[25]Turunen, J. and Wyrowski, F., Diffractive Optics for Industrial and Commercial Applications, Berlin: Akademie, 1997.
[26]Venakides, S., Haider, M. A. and Papanicolaou, V., Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures, SIAM J. Appl. Math., 60 (2000), 16861706.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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