标题： SkelFMM：基于递归骨架化的简化快速多极子方法 显示英文标题

标题： SkelFMM: A Simplified Fast Multipole Method Based on Recursive Skeletonization

摘要： 这项工作介绍了不依赖核的多级算法“skelFMM”，用于评估通过核（如拉普拉斯或赫姆霍兹方程的基本解）连接的$N$点之间的所有成对相互作用。 该方法基于线性代数工具，如随机低秩近似和远场相互作用的“骨架表示”。 这项工作与之前提出的快速多极方法（FMM）的线性代数重表述有关，但其特点是依赖于更简单的数据结构。 特别是，skelFMM不需要“相互作用列表”，因为它在每一层上依赖于近邻之间的代数修改的核相互作用。 与其他不依赖核的算法一样，它只需要核函数的评估，使得该方法能够轻松扩展到二维和三维中的多种不同核。 该算法的简单性使其特别适合在异构硬件架构上进行并行实现。 通过在二维和三维中均匀和非均匀点分布上的数值实验展示了该算法的性能，涉及拉普拉斯和（低频）赫姆霍兹核。 该算法依赖于一个预计算阶段，为给定的点几何构造一个定制的表示。 一旦预计算完成，矩阵-向量乘法通过利用批处理线性代数的GPU加速达到高速度。 显示英文摘要

摘要： This work introduces the kernel-independent multi-level algorithm "skelFMM" for evaluating all pairwise interactions between $N$ points connected through a kernel such as the fundamental solution of the Laplace or the Helmholtz equations. The method is based on linear algebraic tools such as randomized low rank approximation and "skeleton representations" of far-field interactions. The work is related to previously proposed linear algebraic reformulations of the fast multipole method (FMM), but is distinguished by relying on simpler data structures. In particular, skelFMM does not require an "interaction list", as it relies instead on algebraically-modified kernel interactions between near-neighbors at every level. Like other kernel independent algorithms, it only requires evaluation of the kernel function, allowing the methodology to easily be extended to a range of different kernels in 2D and 3D. The simplicity of the algorithm makes it particularly amenable to parallel implementation on heterogeneous hardware architectures. The performance of the algorithm is demonstrated through numerical experiments conducted on uniform and non-uniform point distributions in 2D and 3D, involving Laplace and (low frequency) Helmholtz kernels. The algorithm relies on a precomputation stage that constructs a tailored representation for a given geometry of points. Once the precomputation has completed, the matrix-vector multiplication attains high speed through GPU acceleration that leverages batched linear algebra.