On the numerical stability of sketched GMRES

We perform a backward stability analysis of preconditioned sketched GMRES [Nakatsukasa and Tropp, SIAM J. Matrix Anal. Appl, 2024] for solving linear systems $Ax=b$, and show that the backward stability at iteration $i$ depends on the conditioning of the Krylov basis $B_{1:i}$ as long as the condition number of $A B_{1:i}$ can be bounded by $1/O(u)$, where $u$ is the unit roundoff. Under this condition, we show that sketched GMRES is backward stable as long as the condition number of $B_{1:i}$ is not too large. Under additional assumptions, we then show that the stability of a restarted implementation of sketched GMRES can be independent of the condition number of $B_{1:i}$, and restarted sketched GMRES is backward stable. We also derive sharper bounds that explain why the backward error can be small even in cases when the basis $B_{1:i}$ is very ill-conditioned, which has been observed in the literature but not yet explained theoretically. We present numerical experiments to demonstrate the conclusions of our analysis, and also show that adaptively restarting where appropriate allows us to recover backward stability in sketched GMRES.