Matrix factorizations are among the most important building blocks of scientific computing. State-of-the-art libraries, however, are not communication-optimal, underutilizing current parallel architectures. We present novel algorithms for Cholesky and LU factorizations that utilize an asymptotically communication-optimal 2.5D decomposition. We first establish a theoretical framework for deriving parallel I/O lower bounds for linear algebra kernels, and then utilize its insights to derive Cholesky and LU schedules, both communicating N^3/(P*sqrt(M)) elements per processor, where M is the local memory size. The empirical results match our theoretical analysis: our implementations communicate significantly less than Intel MKL, SLATE, and the asymptotically communication-optimal CANDMC and CAPITAL libraries. Our code outperforms these state-of-the-art libraries in almost all tested scenarios, with matrix sizes ranging from 2,048 to 262,144 on up to 512 CPU nodes of the Piz Daint supercomputer, decreasing the time-to-solution by up to three times. Our code is ScaLAPACK-compatible and available as an open-source library.

中文翻译：

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关于线性代数核的并行 I/O 最优性：近似最优矩阵分解

矩阵分解是科学计算最重要的构建块之一。然而，最先进的库不是通信最佳的，未充分利用当前的并行架构。我们提出了 Cholesky 和 ​​LU 分解的新算法，这些算法利用了渐近通信最优 2.5D 分解。我们首先建立一个理论框架来推导线性代数内核的并行 I/O 下界，然后利用它的见解推导出 Cholesky 和 ​​LU 调度，每个处理器都通信 N^3/(P*sqrt(M)) 个元素，其中M 是本地内存大小。实证结果符合我们的理论分析：我们的实现通信明显少于英特尔 MKL、SLATE 以及渐近通信优化的 CANDMC 和 CAPITAL 库。我们的代码在几乎所有测试场景中都优于这些最先进的库，在 Piz Daint 超级计算机的多达 512 个 CPU 节点上，矩阵大小从 2,048 到 262,144 不等，将解决问题的时间缩短了多达三倍. 我们的代码与 ScaLAPACK 兼容，可作为开源库使用。