Optimizing the Adaptive Fast Multipole Method for Fractal Sets
Optimizing the Adaptive Fast Multipole Method for Fractal Sets
复制标题
优化分形集的自适应快速多极子方法
DOI：
10.1137/140962681
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Eric F Darve
中科院分区：
文献类型：
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作者：
H. Pouransari;Eric F Darve
我们已经进行了详细的分析的快速多极子方法（FMM）的自适应情况下，其中的FMM树的深度是不均匀的。这一领域以前的工作主要集中在特殊类型的自适应分布上，例如，当点在二维流形上积累或在空间中的几个点周围积累时。相反，我们考虑了一种更一般的情况，其中分形集，例如，康托集和推广，用于创建自适应点集。这些集合的特征在于它们的维数，一个介于0和3之间的数字。我们引入了一个数学框架来定义一个收敛的八叉树序列，并在此基础上演示了如何将$N \增加到\infty$。提出了一种新的自适应FMM算法的复杂度分析方法。结果表明，${\cal{O}}（N）$的复杂性是可以实现的任何分布的粒子，当一个修改的自适应FMM利用。我们分析了FMM对分形点分布的性能，以及最佳参数如何能够...
We have performed a detailed analysis of the fast multipole method (FMM) in the adaptive case, in which the depth of the FMM tree is nonuniform. Previous works in this area have focused mostly on special types of adaptive distributions, for example, when points accumulate on a two-dimensional manifold or accumulate around a few points in space. Instead, we considered a more general situation in which fractal sets, e.g., Cantor sets and generalizations, are used to create adaptive sets of points. Such sets are characterized by their dimension, a number between 0 and 3. We introduced a mathematical framework to define a converging sequence of octrees, and based on that, demonstrated how to increase $N \to \infty$. A new complexity analysis for the adaptive FMM is introduced. It is shown that the ${\cal{O}}(N)$ complexity is achievable for any distribution of particles, when a modified adaptive FMM is exploited. We analyzed how the FMM performs for fractal point distributions, and how optimal parameters can ...