Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves

Gavin, Brendan; Polizzi, Eric

doi:10.1002/nla.2188

Cited by 13 publications

(15 citation statements)

References 33 publications

(101 reference statements)

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“…Unlike a contour integral-based FEAST eigensolver, where P S is expressed as a contour integral and is approximated by a numerical quadrature, which needs to solve a sequence of large shifted linear systems whose number is the number of nodes times the subspace dimension at each iteration. In order to ensure that the FEAST solver converges linearly, approximate solutions of the shifted linear systems must have increasing accuracy as outer iterations proceeds [7,31], causing that the method may be very costly. Even worse, the accuracy requirement may not be ensured when a shifted linear system is ill conditioned.…”

Section: Discussionmentioning

confidence: 99%

A FEAST SVDsolver based on Chebyshev--Jackson series for computing partial singular triplets of large matrices

Jia¹,

Zhang²

2022

Preprint

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The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix A with the singular values in a given interval. It is subspace iteration in nature applied to an approximate spectral projector associated with the cross-product matrix A T A and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which A is projected to obtain approximations to the desired singular triplets. Approximate spectral projectors are constructed using the Chebyshev-Jackson series expansion other than contour integration and quadrature rules, and they are proven to be always symmetric positive semi-definite with the eigenvalues in [0, 1]. Compact estimates are established for pointwise approximation errors of a specific step function that corresponds to the exact spectral projector, the accuracy of the approximate spectral projector, the number of desired singular triplets, the distance between the desired right singular subspace and the subspace generated each iteration, and the convergence of the FEAST SVDsolver. Practical selection strategies are proposed for the series degree and the subspace dimension. Numerical experiments illustrate that the FEAST SVDsolver is robust and efficient.

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Section: Discussionmentioning

confidence: 99%

A FEAST SVDsolver based on Chebyshev--Jackson series for computing partial singular triplets of large matrices

Jia¹,

Zhang²

2022

Preprint

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“…Because each linear system can be solved independently from the others, this class of eigensolvers naturally lends itself to multiple layers of parallelism, making contourbased eigensolvers especially well suited for today's increasingly parallel computer architectures. As shown in several recent publications, the performance of the eigensolver depends on the effectiveness of the spectral filter r(A, B) [18,35,12,2,5,9]. Recently, the authors of [36] proposed a numerical optimization approach alternative to the standard quadrature rules.…”

Section: Methodsmentioning

confidence: 99%

“…For the Hermitian eigenproblem Ax = λx with λ ∈ [a, b] ⊂ R, the last decade has seen the emergence of a new class of eigensolvers based on spectral projectors. Such eigensolvers are typically expressed as integrals of the spectral resolvent (A − zI) −1 over a contour in the complex plane that encloses the interval [a, b] [32,33,31,9,21]. Numerical quadrature transforms the contour integral into a matrix-valued rational function with complex coefficients β i and poles z i .…”

mentioning

confidence: 99%

Rational spectral filters with optimal convergence rate

Kollnig,

Bientinesi,

Di Napoli

2020

Preprint

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In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through non-linear least square optimization of so-called rational filters, we introduce a systematic method to design these filters by minimizing the worst-case convergence ratio and eliminate the parametric dependence on weight functions. Further, we provide an efficient way to deal with the box-constraints which play a central role for the use of iterative linear solvers in contour-based eigensolvers. Indeed, these parameter-free filters consistently minimize the number of iterations and the number of FLOPs to reach convergence in the eigensolver. As a byproduct, our rational filters allow for a simple solution to load balancing when the solution of an interior eigenproblem is approached by the slicing of the sought after spectral interval.

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“…We plan to combine this approach with relaxed stopping criteria for solving the linear systems in BEAST-C and -M iteratively; cf. also [9,10] for related work.…”

Section: Changing Precision In Subspace Iteration-based Eigensolversmentioning

confidence: 99%

Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects

Alvermann

Basermann

Bungartz

et al. 2019

Japan J. Indust. Appl. Math.

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We first briefly report on the status and recent achievements of the ELPA-AEO (Eigenvalue Solvers for Petaflop Applications -Algorithmic Extensions and Optimizations) and ESSEX II (Equipping Sparse Solvers for Exascale) projects. In both collaboratory efforts, scientists from the application areas, mathematicians, and computer scientists work together to develop and make available efficient highly parallel methods for the solution of eigenvalue problems. Then we focus on a topic addressed in both projects, the use of mixed precision computations to enhance efficiency. We give a more detailed description of our approaches for benefiting from either lower or higher precision in three selected contexts and of the results thus obtained.Keywords ELPA-AEO · ESSEX · eigensolver · parallel · mixed precision IntroductionEigenvalue computations are at the core of simulations in various application areas, including quantum physics and electronic structure computations. Being able to best utilize the capabilities of current and emerging high-end computing systems is essential for further improving such simulations with respect to space/time resolution or by including additional effects in the models. Given these needs, the ELPA-AEO and ESSEX-II projects contribute to the development and implementation of efficient highly parallel methods for eigenvalue problems, in different contexts.Both projects are aimed at adding new features (concerning, e.g., performance and resilience) to previously developed methods and at providing additional functionality with new methods. Building on the results of the first ESSEX funding phase [14,34], ESSEX-II again focuses on iterative methods for very large eigenproblems arising, e.g., in quantum physics. ELPA-AEO's main application area is electronic structure computation, and for these moderately sized eigenproblems direct methods are often superior. Such methods are available in the widely used ELPA library [19], which had originated in an earlier project [2] and is being improved further and extended with ELPA-AEO.In Sections 2 and 3 we briefly report on the current state and on recent achievement in the two projects, with a focus on aspects that may be of particular interest to prospective users of the software or the underlying methods.Mixed precision in the ELPA-AEO and ESSEX-II projects 3In Section 4 we turn to computations involving different precisions. Looking at three examples from the two projects we describe how lower or higher precision is used to reduce the computing time. The ELPA-AEO projectIn the ELPA-AEO project, chemists, mathematicians and computer scientists from the Max Planck Computing and Data Facility in Garching, the Fritz Haber Institute of the Max Planck Society in Berlin, the Technical University of Munich, and the University of Wuppertal collaborate to provide highly scalable methods for solving moderately-sized (n 10 6 ) Hermitian eigenvalue problems. Such problems arise, e.g., in electronic structure computations, and during the earlier ELPA project, efficient...

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Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves

Cited by 13 publications

References 33 publications

A FEAST SVDsolver based on Chebyshev--Jackson series for computing partial singular triplets of large matrices

A FEAST SVDsolver based on Chebyshev--Jackson series for computing partial singular triplets of large matrices

Rational spectral filters with optimal convergence rate

Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects

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