Parallel preconditioned conjugate gradient optimization of the Rayleigh quotient for the solution of sparse eigenproblems

Bergamaschi, Luca; Martínez, Ángeles; Pini, Giorgio

doi:10.1016/j.amc.2005.09.015

Cited by 12 publications

(23 citation statements)

References 24 publications

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“…The same fate appears to befall the ideas of other authors. The CG algorithms of [6] are mentioned in neither of the recent publications [1,2], and although referenced, this paper has not attracted any comments of [15]. The results of the latter paper, in turn, are not mentioned by [2], who have chosen to implement the algorithm of [26] clearly inferior to that of [15], as can be seen from the numerical tests of the present paper.…”

Section: Introductionmentioning

confidence: 49%

“…2 have visible effects on the performance of the algorithms in focus.A rather erratic convergence behaviour of Algorithm 3.2, in particular, its 'non-CG-like' performance in the first series of Laplacian tests, is likely to be the result of an internal conflict between its two major components, the Rayleigh-Ritz procedure and CG, discussed in Section 2.2.Convergence behaviour of the remaining four algorithms illuminates the importance of 'proper' conjugation of search directions in a block CG algorithm. The trace minimization, block Polak-Ribiére and Jacobi algorithms use the generic block scheme (24) and (25) with the conjugation matrix B i that is a multiple of identity, a diagonal matrix and a full rectangular matrix, respectively.…”

Section: Discussionmentioning

confidence: 97%

“…Figs respectively, and Tables 3-5 show CPU times in seconds. 2 In all tests m ¼ n e þ 5, and in all algorithms we opt for storing Limages in order to reduce the number of multiplications by L. We observe that the successive computation of eigenvalues by Polak-Ribiére algorithm is clearly non-competitive. The rest do reasonably well, with LOBPCG and Jacobi algorithms demonstrating superior (and virtually identical) convergence.…”

Section: Standard Problem Testsmentioning

confidence: 98%

“…2 It should be noted that the number of eigenpairs computed in most tests is less than the number of eigenpairs of practical interest. Nevertheless, the reported results are practically meaningful in the following sense.…”

Section: Generalized Problem Testsmentioning

confidence: 99%

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Computing several eigenpairs of Hermitian problems by conjugate gradient iterations

Ovtchinnikov¹

2008

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 49%

Section: Discussionmentioning

confidence: 97%

Section: Standard Problem Testsmentioning

confidence: 98%

Section: Generalized Problem Testsmentioning

confidence: 99%

See 2 more Smart Citations

Computing several eigenpairs of Hermitian problems by conjugate gradient iterations

Ovtchinnikov¹

2008

Journal of Computational Physics

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“…over all vectors x 2 R n lying in a subspace of size S > s via a preconditioned CG-like procedure. Among the different variants of this technique, we elected to use the simultaneous Rayleigh quotient modified CG (SRQMCG) scheme [5,6]. The latter is a Rayleigh-Ritz method relying upon 798 M. FERRONATO, C. JANNA AND G. PINI the solution of the so-called correction equation to enlarge the search space by using a suitable preconditioned Krylov solver.…”

Section: Introductionmentioning

confidence: 99%

Efficient parallel solution to large‐size sparse eigenproblems with block FSAI preconditioning

Ferronato

Janna

Pini

2011

Numerical Linear Algebra App

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The choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for largesize sparse eigenproblems. Although incomplete factorizations with partial fill-in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large-size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi-Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid-dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. (10) where J i b D fl W m.i b 1/ C 1 6 l 6 mi b g and F OEJ i b , W is the rectangular matrix consisting of all rows with index i 2 J i b . It is easy to show that B i b is SPD [15], so an IC decomposition with 800 M. FERRONATO, C. JANNA AND G. PINI BFSAI-IC outperforms FSAI even by a factor larger than 7, whereas with n p > 32, the ratio R progressively approaches 1. Hence, on massively parallel computers with more than 128 processors, the performance of BFSAI-IC and FSAI proves almost equivalent. This should be no surprise as theoretically the limit of BFSAI for n p ! C1 is FSAI itself [15]. Performance with the PJD algorithmAs is known from literature (e.g., [4,7,10,29]), the PJD algorithm can be very efficient and also quite sensitive to the selected combination of user-specified parameters. After several tests, the most Figure 5. Ratio R between the total wall clock time elapsed using FSAI and BFSAI-IC, versus the number of processors n p for computing the s D 10 (left) and s D 40 (right) leftmost eigenpairs with the PJD algorithm. different methods are stopped when the average relative residual of the leftmost s eigenpairs becomes smaller than 10 5 (10 3 for FAULT-639), that is, the highest accuracy that can be achieved

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Parallel Jacobi–Davidson with block FSAI preconditioning and controlled inner iterations

Ferronato

Janna

Pini

2016

Numerical Linear Algebra App

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The Jacobi–Davidson (JD) algorithm is considered one of the most efficient eigensolvers currently available for non-Hermitian problems. It can be viewed as a coupled inner-outer iteration, where the inner one expands the search subspace and the outer one reduces the eigenpair residual. One of the difficulties in the JD efficient use stems from the definition of the most appropriate inner tolerance, so as to avoid useless extra work and keep the number of outer iterations under control. To this aim, the use of an efficient preconditioner for the inner iterative solver is of paramount importance. The present paper describes a fresh implementation of the JD algorithm with controlled inner iterations and block factorized sparse approximate inverse preconditioning for non-Hermitian eigenproblems in a parallel computational environment. The algorithm performance is investigated by comparison with a freely available software package such as SLEPc. The results show that combining the inner tolerance control with an efficient preconditioning technique can allow for a significant improvement of the JD performance, preserving a good scalability

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Parallel preconditioned conjugate gradient optimization of the Rayleigh quotient for the solution of sparse eigenproblems

Cited by 12 publications

References 24 publications

Computing several eigenpairs of Hermitian problems by conjugate gradient iterations

Computing several eigenpairs of Hermitian problems by conjugate gradient iterations

Efficient parallel solution to large‐size sparse eigenproblems with block FSAI preconditioning

Parallel Jacobi–Davidson with block FSAI preconditioning and controlled inner iterations

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