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：计算f(A)b（矩阵函数作用于向量）是科学计算中许多领域共同面临的任务。在许多应用中，矩阵A是稀疏的，但规模极大以至于只能存储极少数量的Krylov基向量。本文提出一种新方法克服这些限制，该方法将随机草图化技术与f(A)b的积分表示相结合。我们引入了两种不同的近似方案：基于草图化FOM的近似和基于草图化GMRES的近似。针对正实矩阵的Stieltjes函数，分析了后一种方法的收敛性。此外，我们推导了草图化FOM近似的闭式表达式，并界定了其与完整FOM近似的距离。数值实验展示了所提草图化方法的潜力。