Thanks to its great potential in reducing both computational cost and memory requirements, combining sketching and Krylov subspace techniques has attracted a lot of attention in the recent literature on projection methods for linear systems, matrix function approximations, and eigenvalue problems. Applying this appealing strategy in the context of linear matrix equations turns out to be far more involved than a straightforward generalization. These difficulties include establishing well-posedness of the projected problem and deriving possible error estimates depending on the sketching properties. Further computational complications include the lack of a natural residual norm estimate and of an explicit basis for the generated subspace. In this paper we propose a new sketched-and-truncated polynomial Krylov subspace method for Sylvester equations that aims to address all these issues. The potential of our novel approach, in terms of both computational time and storage demand, is illustrated with numerical experiments. Comparisons with a state-of-the-art projection scheme based on rational Krylov subspaces are also included.

翻译：由于其降低计算成本和内存需求的巨大潜力，将草图技术与Krylov子空间技术相结合，在近期关于线性系统投影方法、矩阵函数逼近和特征值问题的文献中引起了广泛关注。将这一颇具吸引力的策略应用于线性矩阵方程时，其复杂性远超直接推广。这些困难包括：建立投影问题的适定性，以及根据草图性质推导可能的误差估计。进一步的计算复杂性还包括：缺乏自然的残差范数估计，以及对生成子空间缺乏显式基。本文针对Sylvester方程提出一种新的草图截断多项式Krylov子空间方法，旨在解决上述所有问题。数值实验展示了我们新方法在计算时间和存储需求方面的潜力。同时，我们还与基于有理Krylov子空间的最新投影方案进行了比较。