SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1290-1318, January 2021.
The computation of matrix functions $f(A)$, or related quantities like their trace, is an important but challenging task, in particular, for large and sparse matrices $A$. In recent years, probing methods have become an often considered tool in this context, as they allow one to replace the computation of $f(A)$ or tr$(f(A))$ by the evaluation of (a small number of) quantities of the form $f(A)v$ or $v^Tf(A)v$, respectively. These quantities can then be efficiently computed by standard techniques like, e.g., Krylov subspace methods. It is well known that probing methods are particularly efficient when $f(A)$ is approximately sparse, e.g., when the entries of $f(A)$ show a strong off-diagonal decay, but a rigorous error analysis is lacking so far. In this paper we develop new theoretical results on the existence of sparse approximations for $f(A)$ and error bounds for probing methods based on graph colorings. As a by-product, by carefully inspecting the proofs of these error bounds, we also gain new insights into when to stop the Krylov iteration used for approximating $f(A)v$ or $v^Tf(A)v$, thus allowing for a practically efficient implementation of the probing methods.

中文翻译：

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衰减矩阵函数稀疏逼近和迹估计的探测技术分析

SIAM 矩阵分析与应用杂志，第 42 卷，第 3 期，第 1290-1318 页，2021 年 1 月。
矩阵函数 $f(A)$ 或相关量（如其迹）的计算是一项重要但具有挑战性的任务，特别是对于大型和稀疏矩阵 $A$。近年来，探测方法已成为这种情况下经常被考虑的工具，因为它们允许人们用（少量的) 分别为 $f(A)v$ 或 $v^Tf(A)v$ 形式的数量。然后可以通过标准技术（例如 Krylov 子空间方法）有效地计算这些量。众所周知，当 $f(A)$ 近似稀疏时，探测方法特别有效，例如，当 $f(A)$ 的条目表现出强烈的非对角线衰减时，但目前缺乏严格的误差分析. 在本文中，我们针对 $f(A)$ 的稀疏近似的存在以及基于图着色的探测方法的误差范围开发了新的理论结果。作为副产品，通过仔细检查这些错误界限的证明，我们还获得了关于何时停止用于逼近 $f(A)v$ 或 $v^Tf(A)v$ 的 Krylov 迭代的新见解，因此允许实际有效地实施探测方法。