High-Performance Solvers for Dense Hermitian Eigenproblems

Petschow, Matthias; Peise, Elmar; Bientinesi, Paolo

doi:10.1137/110848803

Cited by 24 publications

(37 citation statements)

References 47 publications

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“…On the other hand, the algorithm employed for the solution of the tridiagonal problem, (IV), is a rather new parallel implementation of the MRRR algorithm by Petschow and coworkers. [24] A recent detailed benchmark compared components from ScaLAPACK, Elemental, and the ELPA library on the BlueGene/Q architecture [73], including the important aspect of hyperthreading (the possible use of more than one MPI task per CPU core) on that machine. In short, ELPA and Elemental showed the best performance on this architecture, with a slight edge for the two-step reduction approach (IIIa) and (IIIb) as implemented in ELPA.…”

Section: Recent Developments In Parallel Eigenvalue Solversmentioning

confidence: 99%

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

Marek

Blüm

Johanni

et al. 2014

J. Phys.: Condens. Matter

226

254

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Abstract. Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N 3 ) with the size of the investigated problem, N (e.g., the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the ELPA library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the ScaLAPACK library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g., Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPIOpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem sizes arising in the electronic structure theory is demonstrated for current high-performance computer architectures such as Cray or Intel/Infiniband. For a ELPA -Eigenvalue Solutions for Electronic Structure Theory 2 matrix of dimension 260,000, scalability up to 295,000 CPU cores has been shown on BlueGene/P.

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Section: Recent Developments In Parallel Eigenvalue Solversmentioning

confidence: 99%

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

Marek

Blüm

Johanni

et al. 2014

J. Phys.: Condens. Matter

226

254

View full text Add to dashboard Cite

show abstract

“…The study is limited to sequential executions and does not take into account the degree of parallelism the algorithms provide. However, various studies [4,38,33,35,32] of the performance and accuracy of parallel implementations come to similar conclusions.…”

Section: Introductionmentioning

confidence: 80%

“…12 Criterion I is used in LAPACK [12] and in results of mr3smp in [32], which usually uses II. Criterion II is used in ScaLAPACK [38] and Elemental [33]. In massively parallel computing environments, criteria III and IV can (and should) be additionally complemented with the splitting based on absolute gaps; see also [34].…”

Section: Adjusting the Algorithmmentioning

confidence: 99%

Improved Accuracy and Parallelism for MRRR-Based Eigensolvers---A Mixed Precision Approach

Petschow¹,

Quintana–Ort́ı²,

Bientinesi³

2014

SIAM J. Sci. Comput.

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The real symmetric tridiagonal eigenproblem is of outstanding importance in numerical computations; it arises frequently as part of eigensolvers for standard and generalized dense Hermitian eigenproblems that are based on a reduction to real tridiagonal form. For its solution, the algorithm of multiple relatively robust representations (MRRR) is among the fastest methods. Although fast, the solvers based on MRRR do not deliver the same accuracy as competing methods like Divide & Conquer or the QR algorithm. In this paper, we demonstrate that the use of mixed precisions leads to improved accuracy of MRRR-based eigensolvers with limited or no performance penalty. As a result, we obtain eigensolvers that are not only as accurate as or more accurate than the best available methods, but also-under most circumstances-faster and more scalable than the competition.

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“…In addition, the highly scalable parallel implementation of the MRRR algorithm is also proposed [15]. However, for the matrices that have extremely clustered eigenvalues such as the Glued-Wilkinson matrix [16], [17], the execution time for the MRRR algorithm is reported to be longer than the time for the bisection algorithm and the inverse iteration algorithm, even though the MRRR is superior with respect to the computational cost [16].…”

Section: Introductionmentioning

confidence: 99%

A New Parallel Symmetric Tridiagonal Eigensolver Based on Bisection and Inverse Iteration Algorithms for Shared-Memory Multi-core Processors

Ishigami

Kimura

Nakamura

2015

2015 10th International Conference on P2P, Parallel, Grid, Cloud and Internet Computing (3PGCIC)

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In order to accelerate the subset computation of eigenpairs for real symmetric tridiagonal matrices on sharedmemory multi-core processors, a parallel symmetric tridiagonal eigensolver is proposed, which computes eigenvalues of target matrices using the parallel bisection algorithm and computes the corresponding eigenvectors using the block inverse iteration algorithm with reorthogonalization (BIR algorithm). The BIR algorithm is based on the simultaneous inverse iteration (SI) algorithm, which is a variant of the inverse iteration algorithm, and is introduced to a block parameter. Since the BIR algorithm is mainly composed of the matrix multiplications, the proposed eigensolver is expected to accelerate the computation of eigenpairs even on massively parallel computers. Numerical experiments on shared-memory multi-core processors show that the BIR algorithm is faster than the SI algorithm and achieves the good parallel efficiency. In addition, many cases of the numerical experiments also show that the proposed eigensolver, including the parallel bisection and the BIR algorithm, is more accurate than the parallel implementation of other eigensolvers, such as the QR iteration algorithm, the divide-and-conquer algorithm, and the multiple relatively robust representations algorithm.

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High-Performance Solvers for Dense Hermitian Eigenproblems

Cited by 24 publications

References 47 publications

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

Improved Accuracy and Parallelism for MRRR-Based Eigensolvers---A Mixed Precision Approach

A New Parallel Symmetric Tridiagonal Eigensolver Based on Bisection and Inverse Iteration Algorithms for Shared-Memory Multi-core Processors

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