Improving robustness of the FEAST algorithm and solving eigenvalue problems from graphene nanoribbons

Galgon, Martin; Krämer, Lukas; Lang, Bruno; Alvermann, Andreas; Fehske, Holger; Pieper, Andreas

doi:10.1002/pamm.201410391

Cited by 8 publications

(27 citation statements)

References 7 publications

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“…For the sake of this discussion, we consider the shifted matrix A z := zI − A, which is an instance of the system matrix in (2). All considerations made in the following can easily be transferred to the generalized eigenvalue problem with B = I Hermitian and positive definite via a simple transformation [20].…”

Section: Properties Of Complex-shifted Linear Systemsmentioning

confidence: 99%

“…In order to better understand its properties, simulations involving large sparse matrices are necessary, and a comparatively large number of inner eigenpairs is required for instance to compute an electric current through a graphene sample. This application was previously discussed in the context of the ESSEX project [2,18]. In our experiments, graphene samples of W × L atoms are described mathematically, resulting in Hermitian matrices of size WL × WL.…”

Section: Introductionmentioning

confidence: 99%

“…It arises in many application areas such as electronic structure calculations [1] or modeling of graphene nanoribbons [2], see also below. Typically, only a subset of the eigensystem is required, i. e., m < N eigenvalues with corresponding eigenvectors have to be computed.…”

Section: Introductionmentioning

confidence: 99%

“…The solution of very large and sparse linear systems is typically performed iteratively, e. g., with Krylov-based methods [13]. For linear systems (2) as found in the FEAST algorithm, standard methods such as GMRES [14] and other Krylov-based solvers show poor performance [9], requiring hundreds of iterations even for small system sizes. In the original publication on FEAST [8,Example III], GMRES was used to solve the inner linear systems to modest accuracy.…”

Section: Introductionmentioning

confidence: 99%

“…In [18], the distribution of eigenvalues for a typical problem from graphene modeling can be found. The eigenpairs of interest are those with eigenvalue close to the center of the spectrum [2,18]. The basic configuration of graphene is a two-dimensional hexagonal lattice of carbon atoms, and we consider interactions with nearest and next-nearest neighbors.…”

Section: Introductionmentioning

confidence: 99%

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On the parallel iterative solution of linear systems arising in the FEAST algorithm for computing inner eigenvalues

et al. 2015

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a b s t r a c tMethods for the solution of sparse eigenvalue problems that are based on spectral projectors and contour integration have recently attracted more and more attention. Such methods require the solution of many shifted sparse linear systems of full size. In most of the literature concerning these eigenvalue solvers, only few words are said on the solution of the linear systems, but they turn out to be very hard to solve by iterative linear solvers in practice. In this work we identify a row projection method for the solution of the inner linear systems encountered in the FEAST algorithm and introduce a novel hybrid parallel and fully iterative implementation of the eigenvalue solver. Our approach ultimately aims at achieving extreme parallelism by exploiting the algorithm's potential on several levels. We present numerical examples where graphene modeling is one of the target applications. In this application, several hundred or even thousands of eigenvalues from the interior of the spectrum are required, which is a big challenge for state-of-the-art numerical methods.

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Section: Properties Of Complex-shifted Linear Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the parallel iterative solution of linear systems arising in the FEAST algorithm for computing inner eigenvalues

et al. 2015

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Improving projection‐based eigensolvers via adaptive techniques

Galgon

Krämer

Lang

2017

Numerical Linear Algebra App

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SummaryWe consider subspace iteration (or projection-based) algorithms for computing those eigenvalues (and associated eigenvectors) of a Hermitian matrix that lie in a prescribed interval. For the case that the projector is approximated with polynomials, we present an adaptive strategy for selecting the degree of these polynomials such that convergence is achieved with near-to-optimum overall work without detailed a priori knowledge about the eigenvalue distribution.The idea is then transferred to the approximation of the projector by numerical integration, which corresponds to FEAST algorithm proposed by E. Polizzi in 2009. [E. Polizzi: Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 2009; 79:115112]. Here, our adaptation controls the number of integration nodes. We also discuss the interaction of the method with search space reduction methods.

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Flexible subspace iteration with moments for an effective contour integration‐based eigensolver

Huber

Futamura

Galgon

et al. 2022

Numerical Linear Algebra App

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Contour integration schemes are a valuable tool for the solution of difficult interior eigenvalue problems. However, the solution of many large linear systems with multiple right hand sides may prove a prohibitive computational expense. The number of right hand sides, and thus, computational cost may be reduced if the projected subspace is created using multiple moments. In this work, we explore heuristics for the choice and application of moments with respect to various other important parameters in a contour integration scheme. We provide evidence for the expected performance, accuracy, and robustness of various schemes, showing that good heuristic choices can provide a scheme featuring good properties in all three of these measures.

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Improving robustness of the FEAST algorithm and solving eigenvalue problems from graphene nanoribbons

Abstract: We consider the FEAST eigensolver, introduced by Polizzi in 2009 [5]. We describe an improvement concerning the reliability of the algorithm and discuss an application in the solution of eigenvalue problems from graphene modeling.

Cited by 8 publications

References 7 publications

On the parallel iterative solution of linear systems arising in the FEAST algorithm for computing inner eigenvalues

On the parallel iterative solution of linear systems arising in the FEAST algorithm for computing inner eigenvalues

Improving projection‐based eigensolvers via adaptive techniques

Flexible subspace iteration with moments for an effective contour integration‐based eigensolver

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