Among randomized numerical linear algebra strategies, so-called sketching procedures are emerging as effective reduction means to accelerate the computation of Krylov subspace methods for, e.g., the solution of linear systems, eigenvalue computations, and the approximation of matrix functions. While there is plenty of experimental evidence showing that sketched Krylov solvers may dramatically improve performance over standard Krylov methods, many features of these schemes are still unexplored. We derive new theoretical results that allow us to significantly improve our understanding of sketched Krylov methods, and to identify, among several possible equivalent formulations, the most suitable sketched approximations according to their numerical stability properties. These results are also employed to analyze the error of sketched Krylov methods in the approximation of the action of matrix functions, significantly contributing to the theory available in the current literature.

翻译：在随机数值线性代数策略中，所谓的草图化技术正逐渐成为有效的降维手段，用于加速Krylov子空间方法在例如线性系统求解、特征值计算及矩阵函数逼近中的应用。尽管大量实验证据表明，草图化Krylov求解器可能显著提升标准Krylov方法的性能，但这些方案的许多特性仍有待探索。我们推导出新的理论结果，从而显著加深对草图化Krylov方法的理解，并在多种等价表述中，根据数值稳定性特征识别出最合适的草图化逼近。这些结果还用于分析草图化Krylov方法在矩阵函数作用逼近中的误差，为现有文献中的理论体系作出重要贡献。