This article introduces randomized block Gram-Schmidt process (RBGS) for QR decomposition. RBGS extends the single-vector randomized Gram-Schmidt (RGS) algorithm and inherits its key characteristics such as being more efficient and having at least as much stability as any deterministic (block) Gram-Schmidt algorithm. Block algorithms offer superior performance as they are based on BLAS3 matrix-wise operations and reduce communication cost when executed in parallel. Notably, our low-synchronization variant of RBGS can be implemented in a parallel environment using only one global reduction operation between processors per block. Moreover, the block Gram-Schmidt orthogonalization is the key element in the block Arnoldi procedure for the construction of a Krylov basis, which in turn is used in GMRES, FOM and Rayleigh-Ritz methods for the solution of linear systems and clustered eigenvalue problems. In this article, we develop randomized versions of these methods, based on RBGS, and validate them on nontrivial numerical examples.

翻译：本文介绍了用于QR分解的随机分块Gram-Schmidt过程（RBGS）。RBGS扩展了单向量随机Gram-Schmidt（RGS）算法，并继承了其关键特性，例如效率更高且稳定性不低于任何确定性（分块）Gram-Schmidt算法。分块算法基于BLAS3矩阵级运算，具有更优性能，且并行执行时能降低通信开销。值得注意的是，我们的低同步RBGS变体在并行环境中每分块仅需处理器间执行一次全局归约操作。此外，分块Gram-Schmidt正交化是构建Krylov基的块Arnoldi过程的核心要素，该基进一步用于GMRES、FOM及Rayleigh-Ritz方法以求解线性系统与聚类特征值问题。本文基于RBGS开发了这些方法的随机化版本，并通过非平凡数值算例进行了验证。