The computation of f(A)b, the action of a matrix function on a vector, is a task arising in many areas of scientific computing. In many applications, the matrix A is sparse but so large that only a rather small number of Krylov basis vectors can be stored. Here we discuss a new approach to overcome these limitations by randomized sketching combined with an integral representation of f(A)b. Two different approximations are introduced, one based on sketched FOM and another based on sketched GMRES approximation. The convergence of the latter method is analyzed for Stieltjes functions of positive real matrices. We also derive a closed form expression for the sketched FOM approximant and bound its distance to the full FOM approximant. Numerical experiments demonstrate the potential of the presented sketching approaches.

翻译：F(A)b的计算,即矢量矩阵函数的动作,是许多科学计算领域的一项任务。在许多应用中,矩阵A是稀少的,但规模很大,只能储存少量的Krylov基矢量。在这里,我们讨论通过随机绘制草图和f(A)b的综合表示来克服这些限制的新方法。引入了两种不同的近似值,一种基于草图的FOM,另一种基于草图的GMRES近似值。后一种方法的趋同为正真实矩阵的Stieltjes函数分析。我们还为草图的FOM近似值制作了一种封闭式表达方式,将其与整个FOM 相近的距离绑在一起。数字实验显示了所提出的草图方法的潜力。