A posteriori error estimate for computing tr(<i>f</i>(<i>A</i>)) by using the Lanczos method

Chen, Jie; Saad, Yousef

doi:10.1002/nla.2170

Cited by 6 publications

(3 citation statements)

References 42 publications

(119 reference statements)

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“…For the special case of the extended Krylov subspace and Laplace-Stieltjes functions the convergence to f (A)v is indeed monotonic [55], hence the criterion is reliable. This simple criterion can be further elaborated following the results in [14].…”

Section: Applicationsmentioning

confidence: 99%

The Short-term Rational Lanczos Method and Applications

Palitta¹,

Pozza²,

Simoncini³

2021

Preprint

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Rational Krylov subspaces have become a reference tool in dimension reduction procedures for several application problems. When data matrices are symmetric, a short-term recurrence can be used to generate an associated orthonormal basis. In the past this procedure was abandoned because it requires twice the number of linear system solves per iteration than with the classical long-term method. We propose an implementation that allows one to obtain key rational subspace matrices without explicitly storing the whole orthonormal basis, with a moderate computational overhead associated with sparse system solves. Several applications are discussed to illustrate the advantages of the proposed procedure.

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Section: Applicationsmentioning

confidence: 99%

The Short-term Rational Lanczos Method and Applications

Palitta¹,

Pozza²,

Simoncini³

2021

Preprint

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“…conjugate gradients for shifted linear systems [12,13], stochastic trace estimation [14,15], and stochastic Lanczos quadrature [16][17][18] are suggested in the bibliography.…”

Section: Preliminariesmentioning

confidence: 99%

Stochastic Lanczos estimation of genomic variance components for linear mixed-effects models

Border

Becker

2019

Preprint

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Background: Linear mixed-effects models (LMM) are a leading method in conducting genome-wide association studies (GWAS) but require residual maximum likelihood (REML) estimation of variance components, which is computationally demanding. Previous work has reduced the computational burden of variance component estimation by replacing direct matrix operations with iterative and stochastic methods and by employing loose tolerances to limit the number of iterations in the REML optimization procedure. Here, we introduce two novel algorithms, stochastic Lanczos derivative-free REML (SLDF_REML) and Lanczos first-order Monte Carlo REML (L_FOMC_REML), that exploit problem structure via the principle of Krylov subspace shift-invariance to speed computation beyond existing methods. Both novel algorithms only require a single round of computation involving iterative matrix operations, after which their respective objectives can be repeatedly evaluated using vector operations. Further, in contrast to existing stochastic methods, SLDF_REML can exploit precomputed genomic relatedness matrices (GRMs), when available, to further speed computation.Results: Results of numerical experiments are congruent with theory and demonstrate that interpreted-language implementations of both algorithms match or exceed existing compiled-language software packages in speed, accuracy, and flexibility. Conclusions:Both the SLDF_REML and L_FOMC_REML algorithms outperform existing methods for REML estimation of variance components for LMM and are suitable for incorporation into existing GWAS LMM software implementations.

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“…Recent theoretical results in this direction were given in [18]. Lanczos techniques represent another deterministic approximation approach and are investigated [7,10,31], e.g. Without giving details let us just mention that in order to improve their accuracy, deterministic approximation techniques can be combined with the stochastic techniques to be presented in the sequel; see [46], e.g.…”

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confidence: 99%

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

Frommer¹,

Khalil²,

Ramirez‐Hidalgo³

2021

Preprint

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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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A posteriori error estimate for computing tr(f(A)) by using the Lanczos method

Cited by 6 publications

References 42 publications

The Short-term Rational Lanczos Method and Applications

The Short-term Rational Lanczos Method and Applications

Stochastic Lanczos estimation of genomic variance components for linear mixed-effects models

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

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