In this study, a modified multilevel fast multipole algorithm is constructed for investigating large-scale particle systems. The algorithm expands the number of levels of the modified dual-level fast multipole algorithm from dual-level grids to multipole levels by a layer-by-layer correction and recursive calculation. The linear equations on coarse grid are recursively solved by a two-level grid. The single sparse matrix having higher filling rate is decomposed into a set of sparse matrices with much lower filling rate. Subsequent theoretical analysis and examples demonstrate that the total storage space of sparse matrices is significantly reduced, yet efficiency of the algorithm is almost unaffected. The fast multipole method is applied to expedite the matrix–vector multiplications. Complexity analysis demonstrates the algorithm has O(N) operation efficiency and storage complexity for three-dimensional potential model. A potential example with 10 million degrees of freedom is accurately computed via a single laptop with 16GB RAM. Finally, the development process of the modified multilevel fast multipole algorithm is briefly overviewed.

在这项研究中，构造了一种改进的多级快速多极算法，用于研究大规模粒子系统。该算法通过逐层校正和递归计算，将修改后的双层快速多极点算法的级别数从双层网格扩展到多极点级别。粗糙网格上的线性方程由两级网格递归求解。具有较高填充率的单个稀疏矩阵被分解为一组填充率低得多的稀疏矩阵。随后的理论分析和实例证明，稀疏矩阵的总存储空间显着减少，但算法的效率几乎不受影响。快速多极点方法用于加快矩阵-矢量乘法。复杂度分析表明该算法具有三维电势模型的O（N）操作效率和存储复杂性。一台具有16GB RAM的笔记本电脑可以准确地计算出具有1000万自由度的潜在示例。最后，简要概述了改进的多级快速多极点算法的开发过程。