SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A883-A907, January 2021.
This paper presents an adaptive randomized algorithm for computing the butterfly factorization of an $m\times n$ matrix with $m\approx n$ provided that both the matrix and its transpose can be rapidly applied to arbitrary vectors. The resulting factorization is composed of $\mathcal{O}(\log n)$ sparse factors, each containing $\mathcal{O}(n)$ nonzero entries. The factorization can be attained using $\mathcal{O}(n^{3/2}\log n)$ computation and $\mathcal{O}(n\log n)$ memory resources. The proposed algorithm can be implemented in parallel and can apply to matrices with strong or weak admissibility conditions arising from surface integral equation solvers as well as multi-frontal-based finite-difference, finite-element, or finite-volume solvers. A distributed-memory parallel implementation of the algorithm demonstrates excellent scaling behavior.

中文翻译：

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通过随机矩阵向量乘法进行蝴蝶分解

SIAM科学计算杂志，第43卷，第2期，第A883-A907页，2021年1月。
本文提出了一种自适应随机算法，用于计算带有$ m \大约n $的$ m \ times n $矩阵的蝶形分解，前提是该矩阵及其转置都可以快速应用于任意矢量。所得的因式分解由$ \ mathcal {O}（\ log n）$个稀疏因子组成，每个稀疏因子包含$ \ mathcal {O}（n）$个非零条目。可以使用$ \ mathcal {O}（n ^ {3/2} \ log n）$计算和$ \ mathcal {O}（n \ log n）$内存资源来实现分解。所提出的算法可以并行实现，并且可以应用于由表面积分方程求解器以及基于多边的有限差分，有限元或有限体积求解器产生的具有强或弱容许条件的矩阵。该算法的分布式内存并行实现展示了出色的缩放行为。