采用随机向量的分块Krylov方法近似矩阵函数
Approximating matrix functions by block Krylov methods with randomized vectors
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中文总结 AI 辅助
本文提出含初始块向量的随机分块Krylov方法，用于近似大规模矩阵的函数与向量乘积，可减少计算时间及Krylov步数。
中文摘要 AI 辅助
在应用数学的多个领域中，需要计算形如 $f(A)\mathbf{b}$ 的表达式，其中 $A$ 为方阵，$f$ 为函数，$\mathbf{b}$ 为向量。当矩阵 $A$ 规模极大时，直接计算 $f(A)$ 通常不可行，因此常通过依赖 $A$ 和 $\mathbf{b}$ 的Krylov子空间计算估计值来近似 $f(A)\mathbf{b}$，且仅需对小型矩阵计算 $f$。本文探讨多种随机分块Krylov方法在近似 $f(A)\mathbf{b}$ 中的应用。数值算例表明，初始块向量包含 $\mathbf{b}$ 及少量随机生成向量的分块Krylov方法，相比标准Krylov方法，可减少计算时间并降低Krylov步数。
英文摘要
The need to evaluate expressions of the form $f(A)\mathbf{b}$, where $A$ is a square matrix, $f$ is a function, and $\mathbf{b}$ is a vector, arises in several areas of applied mathematics. When the matrix $A$ is very large, it is usually not attractive to evaluate $f(A)$. Instead, $f(A)\mathbf{b}$ often is approximated by computing an estimate in a Krylov subspace that depends on $A$ and $\mathbf{b}$, and only requires that $f$ be evaluated at a small matrix. This paper explores the application of several variants of randomized block Krylov methods to the approximation of $f(A)\mathbf{b}$. Computed examples suggest that block Krylov methods with an initial block vector that contains $\mathbf{b}$ as well as a few randomly generated vectors may require less computing time and reduce the number of Krylov steps than standard Krylov methods.
