论文标题
大规模矩阵函数的Krylov近似值的随机素描
Randomized sketching for Krylov approximations of large-scale matrix functions
论文作者
论文摘要
f（a）b的计算是矩阵函数对向量的作用，是在科学计算的许多领域中产生的任务。在许多应用中，矩阵A稀疏，但如此之大，以至于只能存储仅数量的Krylov基矢量。在这里，我们讨论了一种新方法来克服这些局限性，通过随机素描与F（a）b的积分表示结合在一起。引入了两个不同的近似值，一个基于草图的FOM，另一个基于草图的GMRE近似。分析了阳性真实矩阵的stieltjes函数，分析了后一种方法的收敛性。我们还得出了概述的FOM近似值的封闭形式表达式，并将其距离绑定到完整的FOM近似值。数值实验证明了提出的草图方法的潜力。
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.
