Krylov-Aware Stochastic Trace Estimation

Abstract

.We introduce an algorithm for estimating the trace of a matrix function \(f(\mathbf{A})\) using implicit products with a symmetric matrix \(\mathbf{A}\) . Existing methods for implicit trace estimation of a matrix function tend to treat matrix-vector products with \(f(\mathbf{A})\) as a black box to be computed by a Krylov subspace method. Like other recent algorithms for implicit trace estimation, our approach is based on a combination of deflation and stochastic trace estimation. However, we take a closer look at how products with \(f(\mathbf{A})\) are integrated into these approaches which enables several efficiencies not present in previously studied methods. In particular, we describe a Krylov subspace method for computing a low-rank approximation of a matrix function by a computationally efficient projection onto Krylov subspace.Keywordsspectral functionHutchinson's methodquadratic trace estimationlow-rank approximationblock-Lanczos methodKrylov subspace methodMSC codes15A1665F5065F6068W25

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