Modified Rayleigh Conjecture Method and Its Applications

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 review 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.