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A Multilevel Approach to Variance Reduction in the Stochastic Estimation of the Trace of a Matrix

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arxiv 2108.11281 v1 pith:RKHXUWB3 submitted 2021-08-25 math.NA cs.NAhep-lat

Andreas Frommer , Mostafa Nasr Khalil , Gustavo Ramirez-Hidalgo This is my paper · ORCID
classification math.NAcs.NAhep-lat
keywords tracematrixapproachestimationhigherinversemonte-carlomultilevel
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The trace of a matrix function f(A), most notably of the matrix inverse, can be estimated stochastically using samples< x,f(A)x> if the components of the random vectors x obey an appropriate probability distribution. However such a Monte-Carlo sampling suffers from the fact that the accuracy depends quadratically of the samples to use, thus making higher precision estimation very costly. In this paper we suggest and investigate a multilevel Monte-Carlo approach which uses a multigrid hierarchy to stochastically estimate the trace. This results in a substantial reduction of the variance, so that higher precision can be obtained at much less effort. We illustrate this for the trace of the inverse using three different classes of matrices.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Multigrid low-mode averaging

    hep-lat 2024-12 conditional novelty 6.0 of 10

    A multigrid extension of low-mode averaging keeps the number of Dirac low modes fixed while suppressing stochastic variance on increasingly large lattices.

  2. Variance reduction with probing and Multilevel Monte Carlo in Lattice QCD

    hep-lat 2026-07 conditional novelty 5.0 of 10

    Multigrid MLMC yields up to O(10^5) variance reduction for connected correlators while torus probing with dilution substantially reduces cost for disconnected loops, confirming complementary regimes.

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