论文标题
HODLR2D：新的等级矩阵
HODLR2D: A new class of Hierarchical matrices
论文作者
论文摘要
本文介绍了HODLR2D，这是一种新的层次低级别表示形式，用于在二维中由$ n $身体问题引起的一类密集矩阵。使用这个新的分层框架，我们提出了一种新的快速矩阵矢量产品，该产品几乎线性地缩放。我们将这种快速矩阵向量产物应用于径向基函数插值和离散的积分方程而产生的大型密集线性系统的迭代解决方案。 HODLR2D矩阵向量产品的空间和计算复杂性缩放为$ \ MATHCAL {O}（pn \ log（n））$，其中$ p $是压缩矩阵子块的最大等级。我们还证明了$ p \ in \ Mathcal {o}（\ log（n）\ log（\ log（n）））$，这确保了HODLR2D矩阵-Dector产品的存储和计算复杂性仍然可用于大型$ n $。此外，我们还提供了HODLR2D的并行可伸缩性，作为本文的一部分。
This article introduces HODLR2D, a new hierarchical low-rank representation for a class of dense matrices arising out of $N$ body problems in two dimensions. Using this new hierarchical framework, we propose a new fast matrix-vector product that scales almost linearly. We apply this fast matrix-vector product to accelerate the iterative solution of large dense linear systems arising out of radial basis function interpolation and discretized integral equation. The space and computational complexity of HODLR2D matrix-vector products scales as $\mathcal{O}(pN \log(N))$, where $p$ is the maximum rank of the compressed matrix subblocks. We also prove that $p \in \mathcal{O}(\log(N)\log(\log(N)))$, which ensures that the storage and computational complexity of HODLR2D matrix-vector products remain tractable for large $N$. Additionally, we also present the parallel scalability of HODLR2D as part of this article.
