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Reconstruction of a local perturbation in inhomogeneous periodic layers from partial near field measurements. (English) Zbl 1441.35263

The paper is devoted to electromagnetic scattering on unbounded domains with focus on the inverse problem of recovering a local perturbation in inhomogeneous periodic layers. In particular, the simplified physical model \[ \Delta u+ k^2n^2u=-f \quad \mbox{in} \;\mathbb{R}^d_+\;(d=2,3) \] in the upper half spaces \(\mathbb{R}^2_+\) and \(\mathbb{R}^3_+\) with homogeneous boundary condition is considered. The treatment of the direct problem in Section 2 is based on the Bloch-Floquet transform that allows the authors to prove existence of solutions for the scattering problem. Section 3 analyzes the corresponding inverse scattering problems, where near field measurements of the scattered wave yields the data. The numerical solution scheme and details of reconstruction are provided in Section 4. The use of the regularization scheme CG-REGINN (Regularization based on inexact Newton iteration) introduced and analyzed by A. Rieder [SIAM J. Numer. Anal. 43, No. 2, 604–622 (2005; Zbl 1092.65047)] plays a fundamental role for the procedure. For that, required properties of the forward operator like Fréchet differentiability had been proven in Section 3, where also the local ill-posedness was shown for the associated nonlinear operator equation that models the inverse problem. Section 5 completes the paper by illustrating theory and algorithms with numerical examples and case studies.

MSC:

35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
47J06 Nonlinear ill-posed problems
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization

Citations:

Zbl 1092.65047

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References:

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