本征周期快速多极子方法：逼近最优格林函数
Natively Periodic Fast Multipole Method: Approximating the Optimal Green Function
摘要：
快速多极子方法（FMM）如果使用周期性格林函数来计算每个FMM八叉树节点的相互作用区域中的多极子展开，则“原生地”遵守周期性边界条件。人们可以为这种方法定义“最佳”格林函数，该方法导致在足够高阶的多极子的限制下收敛到等效粒子网格解的数值解。可以推导出最佳格林函数的离散函数方程，但由于其求解方法未知，因此实际上并不有用。相反，本文提出了一个近似的最佳格林函数，是准确的，以优于1 e-3在LMAX范数和1 e-4在L2范数的实际有用的多极计数。这样一个近似最优的格林函数提供了一个实用的方法来实现FMM与周期性的边界条件“本地”，而不需要计算格和或依赖于混合FMM-PM方法。
英文摘要：
The Fast Multipole Method (FMM) obeys periodic boundary conditions "natively" if it uses a periodic Green function for computing the multipole expansion in the interaction zone of each FMM oct-tree node. One can define the "optimal" Green function for such a method that results in the numerical solution that converges to the equivalent Particle-Mesh solution in the limit of sufficiently high order of multipoles. A discrete functional equation for the optimal Green function can be derived, but is not practically useful as methods for its solution are not known. Instead, this paper presents an approximation for the optimal Green function that is accurate to better than 1e-3 in LMAX norm and 1e-4 in L2 norm for practically useful multipole counts. Such an approximately optimal Green function offers a practical way for implementing FMM with periodic boundary conditions "natively", without the need to compute lattice sums or to rely on hybrid FMM-PM approaches.
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