使用随机化的稠密分层半可分矩阵计算的分布式存储器包
A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization
摘要：
我们提出了一个分布式内存库计算密集结构矩阵。一个矩阵被认为是结构化的，如果它的非对角块可以近似为一个秩亏矩阵具有低的数值秩。在这里，我们使用分层半可分离表示（HSS）。这种矩阵出现在许多应用中，例如，有限元法、边界元法等。利用这种结构可以快速求解线性系统和/或快速计算矩阵向量乘积，这是矩阵计算的两个主要组成部分。我们使用的压缩算法，计算输入密集矩阵的HSS形式，依赖于随机采样与一种新的自适应采样机制。讨论了该算法的并行化问题，给出了结构矩阵向量积、结构因子分解的并行化方法和求解程序。该方法的效率在来自不同学术和工业应用的大型问题上得到了证明，最多可达8，000个核心。这项工作是一个更全球性的努力的一部分，STRUMPACK（图形化矩阵包）软件包的计算与稀疏和密集的结构矩阵。因此，尽管这些例程本身很有用，但它们也代表了向分布式内存稀疏求解器方向迈出的一步。
英文摘要：
We present a distributed-memory library for computations with dense structured matrices. A matrix is considered structured if its off-diagonal blocks can be approximated by a rank-deficient matrix with low numerical rank. Here, we use Hierarchically Semi-Separable representations (HSS). Such matrices appear in many applications, e.g., finite element methods, boundary element methods, etc. Exploiting this structure allows for fast solution of linear systems and/or fast computation of matrix-vector products, which are the two main building blocks of matrix computations. The compression algorithm that we use, that computes the HSS form of an input dense matrix, relies on randomized sampling with a novel adaptive sampling mechanism. We discuss the parallelization of this algorithm and also present the parallelization of structured matrix-vector product, structured factorization and solution routines. The efficiency of the approach is demonstrated on large problems from different academic and industrial applications, on up to 8,000 cores. This work is part of a more global effort, the STRUMPACK (STRUctured Matrices PACKage) software package for computations with sparse and dense structured matrices. Hence, although useful on their own right, the routines also represent a step in the direction of a distributed-memory sparse solver.
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