A multilevel approach to stochastic trace estimation

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