关于预处理后的sketched GMRES数值稳定性的分析
On the numerical stability of sketched GMRES
AI总结：
本文分析了预处理后的sketched GMRES在求解线性方程组时的数值稳定性，证明了在特定条件下，sketched GMRES的数值稳定性与Krylov基的条件数有关，并通过实验验证了其稳定性。
摘要：
我们对预处理后的sketched GMRES进行反向稳定性分析，用于求解线性系统$Ax=b$，并证明在迭代次数$i$处的反向稳定性取决于Krylov基$B_{1:i}$的条件数，只要$A B_{1:i}$的条件数可以被$1/O(u)$所限制，其中$u$是单位舍入误差。在此条件下，我们证明只要$B_{1:i}$的条件数不大，sketched GMRES就是反向稳定的。在额外假设下，我们还证明了sketched GMRES的重启实现的稳定性与$B_{1:i}$的条件数无关，并且重启后的sketched GMRES是反向稳定的。我们还推导了更精确的界限，能够更好地捕捉可达到的反向误差，尤其是在$B_{1:i}$非常病态的情况下。我们通过数值实验验证了我们的分析结论，并展示了适当适应性重启可以恢复sketched GMRES的反向稳定性。
英文摘要：
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 better capture the attainable backward error especially for 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
