具有低同步骨架的自适应重启块Krylov子空间方法
Adaptively restarted block Krylov subspace methods with low-synchronization skeletons
摘要：
随着橡树岭国家实验室的Frontier超级计算机最近实现了exascale性能，减少像QR因子分解这样的内核中的通信变得更加迫切。低同步Gram-Schmidt方法首先在[K.作者：J. Ananthan，U. Yang和S.低同步Gram-Schmidt和广义最小残差算法。是林书Alg.应用程序、Vol. 28（2），e2343，2020]，已经被证明可以提高Arnoldi方法在高性能分布式计算中的可扩展性。块版本的低同步的Gram-Schmidt显示出进一步的潜力，加快算法，因为列的高速缓存允许最大限度地利用矩阵矩阵操作。在这项工作中，低同步块Gram-Schmidt变体从[E。Carson，K. Lund，M. Rozložník和S. Thomas，Block Gram-Schmidt算法及其稳定性，Lin. Alg.应用程序、638，pp. 150- 195，2022]变换成块Arnoldi变体，用于块全正交化方法（BFOM）和块广义最小残差方法（BGMRES）。一个自适应的重新启动启发式的发展，以处理不稳定性，出现的增加条件数的Krylov基础。这些方法的性能，准确性和稳定性通过MATLAB编写的灵活的基准测试工具进行评估。该工具的模块性还允许广义块内积，如全球内积。
英文摘要：
With the recent realization of exascale performace by Oak Ridge National Laboratory's Frontier supercomputer, reducing communication in kernels like QR factorization has become even more imperative. Low-synchronization Gram-Schmidt methods, first introduced in [K. Świrydowicz, J. Langou, S. Ananthan, U. Yang, and S. Thomas, Low Synchronization Gram-Schmidt and Generalized Minimum Residual Algorithms, Numer. Lin. Alg. Appl., Vol. 28(2), e2343, 2020], have been shown to improve the scalability of the Arnoldi method in high-performance distributed computing. Block versions of low-synchronization Gram-Schmidt show further potential for speeding up algorithms, as column-batching allows for maximizing cache usage with matrix-matrix operations. In this work, low-synchronization block Gram-Schmidt variants from [E. Carson, K. Lund, M. Rozložník, and S. Thomas, Block Gram-Schmidt algorithms and their stability properties, Lin. Alg. Appl., 638, pp. 150--195, 2022] are transformed into block Arnoldi variants for use in block full orthogonalization methods (BFOM) and block generalized minimal residual methods (BGMRES). An adaptive restarting heuristic is developed to handle instabilities that arise with the increasing condition number of the Krylov basis. The performance, accuracy, and stability of these methods are assessed via a flexible benchmarking tool written in MATLAB. The modularity of the tool additionally permits generalized block inner products, like the global inner product.
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