An estimator for the diagonal of a matrix

Bekas, Costas; Kokiopoulou, Effrosyni; Saad, Yousef

doi:10.1016/j.apnum.2007.01.003

Cited by 163 publications

(193 citation statements)

References 17 publications

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“…[36], whereby eq. (12) holds exactly if each i-th row of V s is orthogonal to all those rows j of V s for which the (ij)-th entry of S is non-vanishing.…”

Section: Probingmentioning

confidence: 99%

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Variance reduction with practical all-to-all lattice propagators

Endress¹,

Peña²,

Sivalingam³

2015

Computer Physics Communications

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Abstract:We discuss all-to-all quark propagator techniques in two (related) contexts within Lattice QCD: the computation of closed quark propagators, and applications to the so-called "eye diagrams" appearing in the computation of non-leptonic kaon decay amplitudes. Combinations of low-mode averaging and diluted stochastic volume sources that yield optimal signal-to-noise ratios for the latter problem are developed. We also apply a recently proposed probing algorithm to compute directly the diagonal of the inverse Dirac operator, and compare its performance with that of stochastic methods. At fixed computational cost the two procedures yield comparable signal-to-noise ratios, but probing has practical advantages which make it a promising tool for a wide range of applications in Lattice

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“…[36], whereby eq. (12) holds exactly if each i-th row of V s is orthogonal to all those rows j of V s for which the (ij)-th entry of S is non-vanishing.…”

Section: Probingmentioning

confidence: 99%

“…A form of graph coloring has been originally exploited to evaluate sparse Jacobian and Hessian matrices [32,33,34,35]. Unlike first attempts based on random number vectors [36], the kind of probing vectors considered in this algorithm is able to exploit the sparsity pattern of S.…”

Section: Probingmentioning

confidence: 99%

Variance reduction with practical all-to-all lattice propagators

Endress¹,

Peña²,

Sivalingam³

2015

Computer Physics Communications

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“…It remains to consider what can be done for much larger systems. Existing "linear scaling methods" may be adapted by adding techniques from linear algebra, such as probing, or some of the techniques seen in [21].…”

Section: Resultsmentioning

confidence: 99%

“…One can ask the question as to what can be done if the pattern is known and we seek only the diagonal. This was explored for the case of banded matrices, in [21]. …”

Section: Eigenvalue-free Methodsmentioning

confidence: 99%

Diagonalization methods in PARSEC

Saad¹,

Zhou

Bekas

et al. 2006

Physica Status Solidi (b)

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The Pseudopotential Algorithm for Real-Space Electronic Calculations (PARSEC) is a software package written by a team consisting of materials scientists and computer scientists over approximately one decade. During this period, the algorithms used in PARSEC have undergone a series of improvements. This paper reviews the nature of the eigenvalue problems encountered in solving the Kohn-Sham equations and discusses the evolution of the diagonalization methods used in PARSEC.

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“…They considered the particular case where T → 0, in which case f reduces to the Heaviside (or step) function. Related works have also considered techniques based on a Chebyshev series expansion of the Heaviside function [1,13]. Other recent works include [4].…”

Section: Introductionmentioning

confidence: 99%

Rational approximation to the Fermi–Dirac function with applications in density functional theory

Sidje

Saad

2010

Numer Algor

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We are interested in computing the Fermi-Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from Density Function Theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We descibe Lanczos methods and sparse direct method to address these systems. Each approach has advanatges and disadvatanges that are illustrated with experiments.

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An estimator for the diagonal of a matrix

Cited by 163 publications

References 17 publications

Variance reduction with practical all-to-all lattice propagators

Variance reduction with practical all-to-all lattice propagators

Diagonalization methods in PARSEC

Rational approximation to the Fermi–Dirac function with applications in density functional theory

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