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：计算矩阵函数 f(A)b 在向量上的作用是科学计算中出现的任务。在许多应用中，矩阵 A 是稀疏的但如此巨大，只有相对较少的 Krylov 基向量可以存储。在这里，我们讨论了一种克服这些限制的新方法，即随机草图法结合 f(A)b 的积分表示。介绍了两种不同的近似方法，一种基于草图 FOM，另一种基于草图 GMRES 近似。分析了后一种方法在正实矩阵的 Stieltjes 函数中的收敛性。我们还推导出了草图 FOM 近似的闭式表达式，并限制其与完整 FOM 近似的距离。数值实验证明了所呈现的草图方法的潜力。