Stabilizing randomized GMRES through flexible GMRES

TL;DR

This paper introduces a robust randomized solver based on flexible GMRES as an outer wrapper for sketched GMRES, offering minimal parameter tuning and non-increasing residual norms.

Contribution

It develops a new residual bound for FGMRES and proposes a practical, efficient randomized solver that enhances stability and robustness in iterative methods.

Findings

The solver maintains non-increasing residual norms.

It requires minimal parameter tuning.

It demonstrates efficiency and robustness in experiments.

Abstract

We explore the use of flexible GMRES as an outer wrapper for sketched GMRES. Building on a new bound for the residual of FGMRES in terms of the residual of the preconditioner, we derive a practical randomized solver that requires very little parameter tuning, while still being efficient and robust in the sense of generating non-increasing residual norms.