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

Frommer, Andreas; Khalil, Mostafa Nasr; Ramirez‐Hidalgo, Gustavo

doi:10.1137/21m1441894

Cited by 6 publications

(27 citation statements)

References 41 publications

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“…This is why the numbers of stochastic samples do not reflect the total arithmetic cost of the methods, in which, in particular, performing the projections has a high cost when the deflating subspace becomes larger. Interestingly, there is no visible dependence on the mass parameter for the MLMC approaches as was already observed in [5].…”

Section: Pos(lattice2022)017supporting

confidence: 56%

“…Numerical computations were performed using Python on a single core of a node with 44 cores Intel(R) Xeon(R) CPU E5-2699 v4 @ 2.20GHz. We demonstrate the benefits of MG-MLMC++ over exactly deflated Hutchinson and the benefits of MG-MLMC with the two types of accuracies by using the Schwinger discretization of the 2-dimensional Dirac operator [16] with the same configuration and parameters as in [5]. In particular, we use 5 different (negative) masses 𝑚 to shift the mass-less Schwinger operator by the respective multiple of the identity, thus yielding operators with increasing condition number.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…For Rademacher vectors, the variance of the Hutchinson estimator for tr(𝐷 −1 ) is given by 1 2 ∥offdiag(𝐷 −1 + 𝐷 −𝑇 ) ∥ 2 𝐹 , for 𝑍 4 -vectors it is ∥offdiag(𝐷 −1 ) ∥ 2 𝐹 ; see [5], e.g., and the heuristics underlying variance reduction techniques typically rely on just reducing ∥𝐷 −1 ∥ 2 𝐹 .…”

Section: Variance Reduction Via Projectionmentioning

confidence: 99%

“…If we have a sparse representation for 𝑈 𝑘 and 𝑉 𝑘 , we can efficiently use very large values for 𝑘 in inexact deflation. This is the case with multigrid prolongation and restriction operators; see [5,7,8] and section 3.…”

Section: Inexact Deflationmentioning

confidence: 99%

“…In [5] we proposed a multilevel Monte Carlo method based on a multigrid hierarchy to reduce the variance. One splits the original matrix 𝐷 −1 1 = 𝐷 −1 into a telescopic sum as:…”

Section: Multigrid Multilevel Monte Carlo (Mg-mlmc) For the Tracementioning

confidence: 99%

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MGMLMC++ as a Variance Reduction Method for Estimating the Trace of a Matrix Inverse

Khalil¹,

Frommer²

2022

Proceedings of the 39th International Symposium on Lattice Field Theory — PoS(LATTICE2022)

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Hutchinson's method estimates the trace of a matrix function 𝑓 (𝐷) stochastically using samples 𝜏 𝐻 𝑓 (𝐷)𝜏, where the components of the random vectors 𝜏 obey an isotropic probability distribution. Estimating the trace of the inverse of a discretized Dirac operator or variants thereof have become a major challenge in lattice QCD simulations, as they represent the disconnected contribution to certain observables. The Hutchinson Monte Carlo sampling, however, suffers from the fact that its accuracy depends quadratically on the sample size, making higher precision estimation very expensive. Meyer, Musco, Musco and Woodruff recently proposed an enhancement of Hutchinson's method, termed Hutch++, in which the sample space is enriched by several vectors of the form 𝑓 (𝐷)𝜁, 𝜁 a random vector as in Hutchinson's method. Theoretical analyses show that under certain circumstances the number of these added sample vectors can be chosen in a way to reduce the dependence of the variance of the resulting estimator from the number 𝑁 of samples from O (1/𝑁) to O (1/𝑁 2 ). In this study we combine Hutch++ with our recently suggested multigrid multilevel Monte Carlo approach. We present results for the Schwinger discretization of the 2-dimensional Dirac operator, revealing that the two approaches contribute additively to variance reduction.

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Section: Pos(lattice2022)017supporting

confidence: 56%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Variance Reduction Via Projectionmentioning

confidence: 99%

Section: Inexact Deflationmentioning

confidence: 99%

“…In [5] we proposed a multilevel Monte Carlo method based on a multigrid hierarchy to reduce the variance. One splits the original matrix 𝐷 −1 1 = 𝐷 −1 into a telescopic sum as:…”

Section: Multigrid Multilevel Monte Carlo (Mg-mlmc) For the Tracementioning

confidence: 99%

See 3 more Smart Citations

MGMLMC++ as a Variance Reduction Method for Estimating the Trace of a Matrix Inverse

Khalil¹,

Frommer²

2022

Proceedings of the 39th International Symposium on Lattice Field Theory — PoS(LATTICE2022)

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Multigrid multilevel Monte Carlo for a lattice with a displacement

Ramirez‐Hidalgo

2023

Proc Appl Math and Mech

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The Hutchinson method can be used to estimate the trace of a function of a matrix A, that is, , stochastically. With the variance of the Hutchinson estimator decreasing as , techniques to reduce the variance of this estimator become necessary due to the high accuracy required by some calculations in scientific computing. In lattice QCD, where A is the Dirac operator D stemming from a discretization of the Dirac equation, the accuracy of some calculations involving renders them practically unfeasible if one uses the Hutchinson estimator to approximate the trace. One such variance reduction technique, known as multigrid multilevel Monte Carlo, has been introduced recently to deal with for the Laplace 2D, Gauge Laplacian, and Schwinger problems. We use it here in the context of lattice QCD, and moreover, we analyze its impact when dealing with , where is the matrix that emerges when a displacement in one of the four spacetime dimensions of the lattice is introduced. This latter trace is of importance when computing generalized parton distributions in lattice QCD.

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