Commun. Comput. Phys., 12 (2012), pp. 1434-1460.


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

Johannes Elschner 1, Guanghui Hu 1*

1 Weierstrass Institute, Mohrenstr. 39, Berlin 10117, Germany.

Received 22 June 2011; Accepted (in revised version) 13 January 2012
Available online 8 May 2012
doi:10.4208/cicp.220611.130112a

Abstract

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 (C^2) 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.

AMS subject classifications: 35R30, 74B05, 78A46, 35Q93

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Key words: Diffraction grating, elastic waves, profile reconstruction, Tikhonov regularization, optimization method.

*Corresponding author.
Email: johannes.elschner@wias-berlin.de (J. Elschner), guanghui.hu@wias-berlin.de (G. Hu)
 

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