Randomized block Gram-Schmidt process for solution of linear systems and eigenvalue problems

Abstract

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.

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