修正瑞利猜想方法及其应用
Modified Rayleigh Conjecture Method and Its Applications
AI总结：
本文综述了修正瑞利猜想（MRC）在二维和三维障碍物散射、静态问题及周期结构散射中的理论基础，并展示了基于MRC的高效数值算法，该方法易于实现且是其他障碍物散射方法的可行替代。
摘要：
瑞利猜想关于表示障碍物$D$外部散射场的级数收敛到边界的结论在工程应用中被广泛使用。然而，该猜想对于某些障碍物是不成立的。AGR引入了修正瑞利猜想（MRC），这是一个精确的数学结果。本文综述了MRC方法在二维和三维障碍物散射问题、静态问题以及周期结构散射中的理论基础。我们还展示了基于MRC的各种散射问题的成功数值算法。MRC方法对于简单和复杂几何形状都易于实现。它被证明是其他障碍物散射方法的可行替代方案。各种直接和逆散射问题需要寻找多个变量函数的全局最小值。稳定性指数方法（SIM）结合了随机和确定性方法来实现这样的最小化。
英文摘要：
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.
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