Restoring similarity in randomized Krylov methods with applications to eigenvalue problems and matrix functions

TL;DR

This paper introduces a modified randomized Arnoldi process that restores similarity with the standard Arnoldi process, ensuring accurate eigenvalue and matrix function approximations while maintaining computational efficiency and robustness.

Contribution

The authors propose a simple modification to the randomized Arnoldi process that enforces similarity with the standard Arnoldi process, improving convergence behavior and approximation accuracy.

Findings

Modified process achieves identical results to standard Arnoldi.

Approach maintains speed of randomized Arnoldi.

Numerical experiments confirm robustness and efficiency.

Abstract

The randomized Arnoldi process has been used in large-scale scientific computing because it produces a well-conditioned basis for the Krylov subspace more quickly than the standard Arnoldi process. However, the resulting Hessenberg matrix is generally not similar to the one produced by the standard Arnoldi process, which can lead to delays or spike-like irregularities in convergence. In this paper, we introduce a modification of the randomized Arnoldi process that restores similarity with the Hessenberg matrix generated by the standard Arnoldi process. This is accomplished by enforcing orthogonality between the last Arnoldi vector and the previously generated subspace, which requires solving only one additional least-squares problem. When applied to eigenvalue problems and matrix function evaluations, the modified randomized Arnoldi process produces approximations that are identical to those obtained with the standard Arnoldi process. Numerical experiments demonstrate that our approach is as fast as the randomized Arnoldi process and as robust as the standard Arnoldi process.