Dissecting the FEAST algorithm for generalized eigenproblems

Krämer, Lukas; Napoli, Edoardo Di; Galgon, Martin; Lang, Bruno; Bientinesi, Paolo

doi:10.1016/j.cam.2012.11.014

Cited by 30 publications

(66 citation statements)

References 12 publications

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“…Nevertheless, in practice |γ p+1 /γ e | = |γ e+1 /γ e | will be close to 1, rendering convergence slow. This slow convergence was observed in Experiment 3.1 of [15].…”

Section: Combining Equations 52 and 53 Givessupporting

confidence: 64%

“…, λ e to almost identically 1. We note that nonconvergence due to p < e was observed in Experiment 3.1 of [15]. …”

Section: Combining Equations 52 and 53 Givesmentioning

confidence: 57%

“…Section 5 analyzes the convergence properties of eigenpairs, taking into account the idiosyncrasies of Q (k) due to the use of ρ(M ) as accelerator. Some of the consequences of these idiosyncrasies were in fact observed in [15], and now have a satisfactory explanation. We also show that eigenvalues of B (k) offer accurate estimates of e, the number of eigenvalues inside I.…”

mentioning

confidence: 76%

“…The n eigenvalues are generated as follows. We pick four eigenvalues in [15,17] by picking three randomly with uniform distribution in the region [15.2, 16.8]. The fourth is set to be 17.…”

mentioning

confidence: 99%

“…j = 1, 2, 3, and |ρ(λ 4 )| = 1/2. These 4 eigenvalues are the only ones in [15,17] and hence e = 4. Next, five eigenvalues are set to be in the interval (17,18] such that the values of |ρ(λ j )| are 2 −ℓ for ℓ = 3, 5, 7, 9, 11.…”

mentioning

confidence: 99%

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FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection

Tang¹,

Polizzi²

2014

SIAM J. Matrix Anal. & Appl.

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Abstract.The calculation of a segment of eigenvalues and their corresponding eigenvectors of a Hermitian matrix or matrix pencil has many applications. A new density-matrix-based algorithm has been proposed recently and a software package FEAST has been developed. The density-matrix approach allows FEAST's implementation to exploit a key strength of modern computer architectures, namely, multiple levels of parallelism. Consequently, the software package has been well received, especially in the electronic structure community. Nevertheless, theoretical analysis of FEAST has lagged. For instance, the FEAST algorithm has not been proven to converge. This paper offers a detailed numerical analysis of FEAST. In particular, we show that the FEAST algorithm can be understood as an accelerated subspace iteration algorithm in conjunction with the Rayleigh-Ritz procedure. The novelty of FEAST lies in its accelerator which is a rational matrix function that approximates the spectral projector onto the eigenspace in question. Analysis of the numerical nature of this approximate spectral projector and the resulting subspaces generated in the FEAST algorithm establishes the algorithm's convergence. This paper shows that FEAST is resilient against rounding errors and establishes properties that can be leveraged to enhance the algorithm's robustness. Finally, we propose an extension of FEAST to handle non-Hermitian problems and suggest some future research directions.Key words. generalized eigenvalue problem, subspace iteration, spectral projection AMS subject classifications. 15A18, 65F151. Introduction. Solving matrix eigenvalue problems is crucial in many scientific and engineering applications. Robust solvers for problems of moderate size are well developed and widely available [2]. These are sometimes referred to as direct solvers [6]. Direct solvers typically calculate the entire spectrum of the matrix or matrix pencil in question. Yet in many applications, especially for those where the underlying linear systems are large and sparse, it is often the case that only selected segments of the spectrum are of interest. Polizzi recently proposed a densitymatrix-based algorithm [21] named FEAST for Hermitian eigenproblems of this kind. From an implementation point of view, FEAST's main building block is a numericalquadrature computation, consisting of solving independent linear systems, each for multiple right hand sides. This building block contains multiple levels of parallelism and thus exploits the features of modern computing architectures very well. A software package FEAST [22] based on this approach has been made available since 2009. Nevertheless, theoretical analysis of FEAST has been lagging its software development. In particular, there is no theoretical study available on the conditions under which FEAST converges, and if so, at what rate. This paper shows that the FEAST algorithm can be understood as a standard subspace iteration in conjunction with the Rayleigh-Ritz procedure. FEAST therefore belongs to the clas...

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“…Nevertheless, in practice |γ p+1 /γ e | = |γ e+1 /γ e | will be close to 1, rendering convergence slow. This slow convergence was observed in Experiment 3.1 of [15].…”

Section: Combining Equations 52 and 53 Givessupporting

confidence: 64%

“…, λ e to almost identically 1. We note that nonconvergence due to p < e was observed in Experiment 3.1 of [15]. …”

Section: Combining Equations 52 and 53 Givesmentioning

confidence: 57%

mentioning

confidence: 76%

“…The n eigenvalues are generated as follows. We pick four eigenvalues in [15,17] by picking three randomly with uniform distribution in the region [15.2, 16.8]. The fourth is set to be 17.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection

Tang¹,

Polizzi²

2014

SIAM J. Matrix Anal. & Appl.

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Efficient estimation of eigenvalue counts in an interval

Napoli

Polizzi

Saad

2016

Numerical Linear Algebra App

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112

142

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Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications, and it is a prerequisite of eigensolvers based on a divide-andconquer paradigm. Often, an exact count is not necessary, and methods based on stochastic estimates can be utilized to yield rough approximations. This paper examines a number of techniques tailored to this specific task. It reviews standard approaches and explores new ones based on polynomial and rational approximation filtering combined with a stochastic procedure. We also discuss how the latter method is particularly wellsuited for the FEAST eigensolver.ESTIMATING EIGENVALUE COUNTS 675 count OEa; b in OEa; b. While this method yields an exact count, it requires two complete LDL T factorizations, and this can be quite expensive for realistic eigenproblems.This paper discusses two alternative methods that provide only an estimate for OEa; b , but which are relatively inexpensive. Both methods work by estimating the trace of the spectral projector P associated with the eigenvalues inside the interval OEa; b. This spectral projector is expanded in two different ways, and its trace is computed by resorting to stochastic trace estimators, for example, [10,11]. The first method utilizes filtering techniques based on Chebyshev polynomials. The resulting projector is expanded as a polynomial function of A. In the second method, the projector is constructed by integrating the resolvent of the eigenproblem along a contour in the complex plane enclosing the interval OEa; b. In this case, the projector is approximated by a rational function of A.For each of the aforementioned methods, we present various implementations depending on the nature of the eigenproblem (generalized versus standard) and cost considerations. Thus, in the polynomial expansion case, we propose a barrier-type filter when dealing with a standard eigenproblem and two high/low pass filters in the case of generalized eigenproblems. In the rational expansion case, we have the choice of using an LU factorization or a Krylov subspace method to solve linear systems. The optimal implementation of each method used for the eigenvalue count depends on the situation at hand and involves compromises between cost and accuracy. While it is not the aim of this paper to explore detailed analysis of these techniques, we will discuss various possibilities and provide illustrative examples.The polynomial and rational expansion methods are motivated by two distinct approaches recently suggested in the context of electronic structure calculations: (i) spectrum slicing and (ii) Cauchy integral eigen-projection. In the spectrum slicing techniques [1], the eigenpairs are computed by dividing the spectrum in many small subintervals, called 'slices' or 'windows'. For each window, a barrier function is approximated by Chebyshev-Jackson polynomials in order to select only the portion of the spectrum in the slice. In this method, it is important to determine an approximate c...

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A FEAST algorithm with oblique projection for generalized eigenvalue problems

Yin

Chan

Yeung

2017

Numer. Linear Algebra Appl.

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Abstract. The contour-integral based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best-known members are the Sakurai-Sugiura (SS) method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non-Hermitian problems. In this paper, we extend the FEAST algorithm to non-Hermitian problems. The approach can be summarized as follows: (i) to construct a particular contour integral to form a subspace containing the desired eigenspace, and (ii) to use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. We also address some implementation issues such as how to choose a suitable starting matrix and design good stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.

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Dissecting the FEAST algorithm for generalized eigenproblems

Cited by 30 publications

References 12 publications

FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection

FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection

Efficient estimation of eigenvalue counts in an interval

A FEAST algorithm with oblique projection for generalized eigenvalue problems

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