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 a new sketched Arnoldi-type relation that allows us to obtain several different new theoretical results. These lead to an improvement of our understanding of sketched Krylov methods, in particular by explaining why the frequently occurring sketched Ritz values far outside the spectral region of A do not negatively influence the convergence of sketched Krylov methods for f (A)b. Our findings also help 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，所谓的草图化方法正成为一种有效的降维手段，用于加速Krylov子空间方法的计算，例如线性方程组求解、特征值计算以及矩阵函数逼近。尽管大量实验证据表明草图化Krylov求解器相比标准Krylov方法能显著提升性能，这些方案的许多特性仍未得到充分探索。我们推导出一种新的草图化Arnoldi型关系式，借此获得了若干不同的新理论结果。这些结果深化了我们对草图化Krylov方法的理解，特别是解释了为何频繁出现的、远超出A谱域的草图化Ritz值不会对f(A)b的草图化Krylov方法收敛性产生负面影响。我们的发现还有助于从多种可能的等价形式中，根据数值稳定性特征识别出最适宜的草图化逼近格式。这些结果亦被用于分析草图化Krylov方法在逼近矩阵函数作用时的误差，为当前文献中的理论体系作出了重要贡献。