Randomized orthogonalization and Krylov subspace methods: principles and algorithms
Jean-Guillaume de Damas, Laura Grigori, Igor Simunec, Edouard Timsit
TL;DR
This work surveys randomized orthogonalization techniques to construct well-conditioned, sketch-orthonormal bases that reduce the cost and communication of Krylov subspace methods while maintaining numerical stability. It introduces the blind-embedding framework (ε-embeddings and OSEs) and contrasts explicit randomized Gram–Schmidt and Householder QR with implicit whitening approaches, highlighting their impact on linear systems, eigenvalue problems, matrix functions, and matrix equations. Theoretical guarantees show quasi-optimality under embedding distortions and practical gains in parallel and mixed-precision contexts, though questions remain about symmetry preservation and robust library support. Overall, randomized orthogonalization offers a versatile toolkit for scalable, stable Krylov methods with significant potential for future standardization and optimization.
Abstract
We present an overview of randomized orthogonalization techniques that construct a well-conditioned basis whose sketch is orthonormal. Randomized orthogonalization has recently emerged as a powerful paradigm for reducing the computational and communication cost of state-of-the-art orthogonalization procedures on parallel architectures, while preserving, and in some cases improving, their numerical stability. This approach can be employed within Krylov subspace methods to mitigate the cost of orthogonalization, yielding a randomized Arnoldi relation. We review the main variants of the randomized Gram--Schmidt and Householder QR algorithms, and discuss their application to Krylov methods for the solution of large-scale linear algebra problems, such as linear systems of equations, eigenvalue problems, the evaluation of matrix functions, and matrix equations.