This article presents a randomized matrix-free method for approximating the trace of $f({\bf A})$, where ${\bf A}$ is a large symmetric matrix and $f$ is a function analytic in a closed interval containing the eigenvalues of ${\bf A}$. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index, and triangle counting in graphs. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators.

翻译：本条提供了一种随机化的无基体方法,以接近 $f( {bf A}), 美元为大对称矩阵, 美元为在封闭间隔内的函数分析, 包含 $_bf A} 美元。 我们的方法使用一种混合的随机化跟踪估计( 即 Hutchinson 的方法)、 Chebyshev 近似和多级 Monte Carlo 技术。 我们通过将哈钦森 方法的现有误差扩展至多级跟踪测算器, 从而确定了这种方法近似误差的大致界限。 数字实验是针对对日志- 定义、 核规范 和 Extrada 指数 以及图形三角数等常见应用进行的。 我们发现, 使用多级技术可以大幅降低现有单级测算器的差异 。