Randomized orthogonalization and Krylov subspace methods: principles and algorithms

TL;DR

This paper reviews randomized orthogonalization techniques that improve the efficiency and stability of Krylov subspace methods for large-scale linear algebra problems by reducing orthogonalization costs.

Contribution

It provides a comprehensive overview of randomized orthogonalization algorithms and their integration into Krylov methods for large-scale linear algebra applications.

Findings

Randomized orthogonalization reduces computational costs.

Improves numerical stability of Krylov subspace methods.

Applicable to large-scale linear systems and eigenvalue problems.

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.