Modified Rayleigh conjecture method and its applications

Abstract

The Rayleigh conjecture about convergence up to the boundary of the series representing the scattered field in the exterior of an obstacle D is widely used by engineers in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), which is an exact mathematical result. In this paper we present the theoretical basis for the MRC method for 2D and 3D obstacle scattering problems, for static problems, and for scattering by periodic structures. We also present successful numerical algorithms based on the MRC for various scattering problems. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. Various direct and inverse scattering problems require finding global minima of functions of several variables. The Stability Index Method (SIM) combines stochastic and deterministic method to accomplish such a minimization.

On the numerical stability of the least-squares method for the planar scattering by obstacles

“…In particular, it implies the convergence in the L 2 norm on the boundary Γ of the scatterer. The following theorem, proved in [10], shows that the convergence on the boundary of the scatterer is sufficient to ensure convergence outside:…”

“…In this article, we investigate the effect of the choice of the quadrature points on the numerical stability of the least-squares method in the particular case of the scattering by an obstacle. The quadrature used to estimate the error is generally either left unspecified [1,2,10,11], or uses general purpose schemes (Chebyshev nodes in [12] or Clenshaw-Curtis rule in [5]). However, as was shown in [8] for the collocation method, the choice of the quadrature point is critical for the stability of the numerical methods, and depends on the shape of the scatterer.…”