We present a new Krylov subspace recycling method for solving a linear system of equations, or a sequence of slowly changing linear systems. Our new method, named GMRES-SDR, combines randomized sketching and deflated restarting in a way that avoids orthogononalizing a full Krylov basis. We provide new theory which characterizes sketched GMRES with and without augmentation as a projection method using a semi-inner product. We present results of numerical experiments demonstrating the effectiveness of GMRES-SDR over competitor methods such as GMRES-DR and GCRO-DR.

翻译：我们提出了一种新的Krylov子空间循环利用方法，用于求解线性方程组或一组缓慢变化的线性系统。新方法命名为GMRES-SDR，通过避免对完整Krylov基进行正交化处理，将随机草图技术与收缩重启策略有机结合。我们建立了新的理论框架，将带增广与不带增广的草图化GMRES表征为采用半内积的投影方法。数值实验结果表明，相较于GMRES-DR和GCRO-DR等竞争方法，GMRES-SDR在有效性方面具有显著优势。