使用 Chunks and Tasks 编程模型的逆因式分解并行化与可扩展性分析
Parallelization and scalability analysis of inverse factorization using the Chunks and Tasks programming model
AI总结：
针对大规模电子结构计算中块稀疏矩阵的逆因式分解需求，提出三种基于 Chunks and Tasks 模型的分布式并行方法，实验证明其计算成本随系统规模线性增长，且局部逆因式分解方法在弱扩展性上优于现有实现且通信开销显著降低。
摘要：
我们提出了三种用于块稀疏 Hermitian 正定矩阵的分布式内存并行逆因式分解方法。这三种方法分别是 AINV 逆 Cholesky 算法的递归变体、迭代细化以及局部逆因式分解。所有这三种方法均使用 Chunks and Tasks 编程模型实现，基于公开可用的 Chunks and Tasks Matrix Library (CHTML) 中的分布式稀疏四叉树矩阵表示和并行矩阵-矩阵乘法。尽管这些算法具有通用性，但这项工作主要动机是大规模电子结构计算中对基组重叠矩阵进行高效且可扩展的逆因式分解的需求。我们对准线性谷氨酸-丙氨酸分子和使用标准高斯基组 STO-3G 离散化的三维水团簇的重叠矩阵进行了各种计算测试，基函数数量超过 1000 万。我们表明，对于此类矩阵，所有三种方法的计算成本仅随系统规模线性增加。我们在理论和数值实验中均表明，在系统规模与进程数成正比增加的弱扩展测试中，基于迭代细化和局部逆因式分解的方法优于先前的并行实现。我们还表明，与基于纯迭代细化的方法相比，局部逆因式分解所需的通信量要少得多。
英文摘要：
We present three methods for distributed memory parallel inverse factorization of block-sparse Hermitian positive definite matrices. The three methods are a recursive variant of the AINV inverse Cholesky algorithm, iterative refinement, and localized inverse factorization, respectively. All three methods are implemented using the Chunks and Tasks programming model, building on the distributed sparse quad-tree matrix representation and parallel matrix-matrix multiplication in the publicly available Chunks and Tasks Matrix Library (CHTML). Although the algorithms are generally applicable, this work was mainly motivated by the need for efficient and scalable inverse factorization of the basis set overlap matrix in large scale electronic structure calculations. We perform various computational tests on overlap matrices for quasi-linear Glutamic Acid-Alanine molecules and three-dimensional water clusters discretized using the standard Gaussian basis set STO-3G with up to more than 10 million basis functions. We show that for such matrices the computational cost increases only linearly with system size for all the three methods. We show both theoretically and in numerical experiments that the methods based on iterative refinement and localized inverse factorization outperform previous parallel implementations in weak scaling tests where the system size is increased in direct proportion to the number of processes. We show also that compared to the method based on pure iterative refinement the localized inverse factorization requires much less communication.
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