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

Application of the second domain derivative in inverse electromagnetic scattering

“…All of these led to comparable results as in the presented examples. Additionally, similar to other iterative approaches in acoustic or electromagnetic inverse scattering the Halley method confirmed the known observation that there is some improvement if reconstructions from some far field patterns generated for instance by three or four incident directions are averaged in each iteration step (see [11,15,21]).…”

“…Furthermore, in case of acoustic scattering problems several numerical implementations are documented (see [6,11] for three dimensional examples). Presumably due to the computational effort, there are only a few results for the full vector valued electromagnetic inverse scattering problem (see [9,15,21]). These approaches are based on boundary integral equations for the electromagnetic scattering problem and the first domain derivative.…”

Topological Imaging Methods for the Iterative Detection of Multiple Impedance Obstacles

“…and deduce that u ε behaves as (1). We obtain then that (30) is also valid in 2D, and since (32), and therefore (25) holds. To show (26), we apply the tangential gradient operator to the boundary integral equation (7) and use that…”

“…In cases where the method stagnates, we propose to use our approximation for some other iterative method to further minimize the cost functional. For instance, it could be used in combination with parametric derivatives through iteratively regularized Gauss-Newton methods (IRGNM) [11,30,32]. Roughly speaking, the method consists in computing first the TD to have an initial guess for the IRGNM.…”

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

“…We start by recalling that for all boldx∈double-struckRd\normalsnormalunormalpnormalp(pinc), any incident field u inc satisfying the Helmholtz equation with wavenumber κ can be written for any boldz∈double-struckRd such thatas follows (see Eqs. (1) and (2) in Chew, 1992):In double-struckR3,where (αn,j(1)) is a complex-valued sequence depending on x but independent on z, and for ɛ > 0 and bold-italicξ∈double-struckS2, we haveHere j n and Y n , j are the spherical Bessel functions of the first kind (see Section 10.1 in Abramowitz and Stegun, 1964) and the spherical harmonic functions (see Appendix B in Le Louër, 2018), respectively.In double-struckR2,where (αn(1)) is a complex-valued sequence depending on x but independent on z andHere J n is the Bessel function of the first kind and order n (see Section 9.1 in Abramowitz and Stegun, 1964).…”