A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles

Abstract

We consider fast and accurate solution methods for the direct and inverse scattering problems by a few three dimensional piecewise homogeneous dielectric obstacles around the resonance region. The forward problem is reduced to a system of second kind boundary integral equations. For the numerical solution of these coupled integral equations we modify a fast and accurate spectral algorithm, proposed by Ganesh and Hawkins [doi:10.1016/j.jcp.2008.01.016], by transporting these equations onto the unit sphere using the Piola transform of the boundary parametrisations. The computational performances of the forward solver are demonstrated on numerical examples for a variety of three-dimensional smooth and non smooth obstacles. The algorithm, that requires the knowledge of the boundary parametrisation and leads to invert small linear systems, is well-suited for the use of geometric optimisation tools to solve the inverse problem of recovering the shape of scatterers from the knowledge of noisy data. Computational details for the application of the iteratively regularised Gauss--Newton method to the numerical solution of the electromagnetic inverse problem are presented. Numerical experiments for the shape detection of multiple obstacles using incomplete radiation pattern data from back and front side are provided. The results in this article can also be applied for solving shape optimisation problems relying on time-harmonic electromagnetic waves. References

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