We present a new Krylov subspace recycling method for solving a linear system of equations, or a sequence of slowly changing linear systems. Our approach is to reduce the computational overhead of recycling techniques while still benefiting from the acceleration afforded by such techniques. As such, this method augments an unprojected Krylov subspace. Furthermore, it combines randomized sketching and deflated restarting in a way that avoids orthogononalizing a full Krylov basis. We call this new method GMRES-SDR (sketched deflated restarting). With this new method, we provide new theory, which initially characterizes unaugmented sketched GMRES as a projection method for which the projectors involve the sketching operator. We demonstrate that sketched GMRES and its sibling method sketched FOM are an MR/OR pairing, just like GMRES and FOM. We furthermore obtain residual convergence estimates. Building on this, we characterize GMRES-SDR also in terms of sketching-based projectors. Compression of the augmented Krylov subspace for recycling is performed using a sketched version of harmonic Ritz vectors. We present results of numerical experiments demonstrating the effectiveness of GMRES-SDR over competitor methods such as GMRES-DR and GCRO-DR.

翻译：本文提出了一种新的Krylov子空间循环方法，用于求解线性方程组或缓慢变化的线性方程组序列。该方法旨在降低循环技术的计算开销，同时仍能利用此类技术提供的加速效果。具体而言，该方法通过扩展未投影的Krylov子空间实现，并结合了随机草图技术与收缩重启策略，从而避免对完整Krylov基进行正交化。我们将这一新方法命名为GMRES-SDR（基于草图的收缩重启方法）。针对该方法，我们建立了新的理论框架：首先将未扩展的草图化GMRES表征为一种投影方法，其投影算子涉及草图算子；证明了草图化GMRES及其姊妹方法草图化FOM构成类似GMRES与FOM的MR/OR配对关系，并获得了残差收敛估计。在此基础上，我们进一步基于草图投影算子对GMRES-SDR进行理论刻画。在循环过程中，扩展Krylov子空间的压缩通过草图化调和Ritz向量实现。数值实验结果表明，相较于GMRES-DR和GCRO-DR等竞争方法，GMRES-SDR具有显著优势。