通信最优的并行和串行Cholesky分解
Communication-optimal Parallel and Sequential Cholesky Decomposition
AI总结：
针对Cholesky分解，通过扩展矩阵乘法的通信下界并设计算法，实现串行和并行环境下通信最优（包括带宽和延迟）的O(n^3)实现。
摘要：
数值算法有两种成本：算术成本和通信成本，其中通信成本指在串行情况下数据在内存层次结构不同层级之间的移动，或在并行情况下数据在连接处理器的网络上的传输。通信成本通常主导算术成本，因此设计最小化通信的算法具有重要意义。本文首先将传统O(n^3)矩阵乘法的通信成本（包括带宽和延迟）已知下界扩展到Cholesky分解，该分解用于求解稠密对称正定线性系统。其次，我们将各种Cholesky分解实现的成本与这些下界进行比较，并确定达到下界的算法和数据结构。在串行情况下，我们考虑了二级和层次内存模型。结合[13,14,15]的先前结果，这为稠密线性代数的三种基本分解（带主元的LU、QR和Cholesky）的O(n^3)实现提供了一组通信最优算法。但本文超越了先前关于串行LU的工作，通过优化任意层级内存层次结构的通信来实现。
英文摘要：
Numerical algorithms have two kinds of costs: arithmetic and communication, by which we mean either moving data between levels of a memory hierarchy (in the sequential case) or over a network connecting processors (in the parallel case). Communication costs often dominate arithmetic costs, so it is of interest to design algorithms minimizing communication. In this paper we first extend known lower bounds on the communication cost (both for bandwidth and for latency) of conventional (O(n^3)) matrix multiplication to Cholesky factorization, which is used for solving dense symmetric positive definite linear systems. Second, we compare the costs of various Cholesky decomposition implementations to these lower bounds and identify the algorithms and data structures that attain them. In the sequential case, we consider both the two-level and hierarchical memory models. Combined with prior results in [13, 14, 15], this gives a set of communication-optimal algorithms for O(n^3) implementations of the three basic factorizations of dense linear algebra: LU with pivoting, QR and Cholesky. But it goes beyond this prior work on sequential LU by optimizing communication for any number of levels of memory hierarchy.
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