Parallel numerical linear algebra

Demmel, James; Heath, Michael T.; Vorst, Henk A. van der

doi:10.1017/s096249290000235x

Cited by 157 publications

(112 citation statements)

References 163 publications

Supporting

Mentioning

110

Contrasting

Unclassified

Order By: Relevance

“…As was established, e.g., in the early ScaLAPACK related literature, [7,75] twodimensional (block) cyclic layouts are crucial for improving scalability up to higher numbers of processors through well-balanced work load and efficient communication patterns. Choosing this layout has the additional advantage that it is easy to insert ELPA into existing code that is already set up for ScaLAPACK.…”

Section: Data Layout and Basic Setupmentioning

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

250

261

View full text Add to dashboard Cite

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.

show abstract

Section: Data Layout and Basic Setupmentioning

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

250

261

View full text Add to dashboard Cite

show abstract

“…A way to get around this difficulty is to create a Krylov subspace basis first and orthogonalize the basis afterwards, see e.g. (2). A similar procedure has been applied in the s-step Arnoldi algorithm described by Kim and Chronopoulos [4).…”

Section: Parallel Implementationmentioning

confidence: 99%

“…Meanwhile considerable progress has been made in the techniques to implement numerical algorithms for large eigenvalue problems on parallel computers. For a recent review in this area, see [2].…”

Section: Introductionmentioning

confidence: 99%

“…The most expensive part in the shift-and-invert Arnoldi method [9] is inverting the shifted matrix: first one has to perform an LU-decomposition of the shifted matrix, followed by solving lower and upper triangular systems of equations. Furthermore, the latter operations make this method unfavourable for an efficient implementation on a distributed memory computer [2].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Parallel Arnoldi method for the construction of a Krylov subspace basis: An application in magnetohydrodynamics

Booten

Meijer

Riele

et al. 1994

High-Performance Computing and Networking

View full text Add to dashboard Cite

Abstract. In a recent article [6] a new method was proposed for computing internal eigenvalues of symmetric matrices. In the present paper we extend these ideas to non-hermitian eigenvalue problems and apply them to a practical example· from the field of magnetohydrodynamics (MHD). The method is very suitable for an efficient parallel implementation. We give some results for the time-consuming kernels of the underlying orthogonalization process, the Arnoldi method, obtained for an MHD problem on a distributed memory multiprocessor.

show abstract

“…Further many researchers have proposed and discussed parallel issues of conjugate gradient methods [2,6,7,8,9,10,21,22]. Additionally sparse approximate inverses by minimizing the Frobenious norm of the error have been presented and can be implemented on parallel systems [17,21].…”

Section: Introductionmentioning

confidence: 99%

On the performance of parallel normalized explicit preconditioned conjugate gradient type methods

Gravvanis

Giannoutakis

2006

Proceedings 20th IEEE International Parallel &Amp; Distributed Processing Symposium

View full text Add to dashboard Cite

A new class of parallel normalized preconditioned conjugate gradient type methods in conjunction with normalized approximate inverses algorithms, based on normalized approximate factorization procedures, for solving sparse linear systems of irregular structure, which are derived from the finite element method of a two dimensional boundary value problem, is introduced. Parallel normalized explicit preconditioned conjugate gradient -type methods for distributed memory systems based on the block -row distribution (for the vectors and the explicit approximate inverse), using Message Passing Interface (MPI) communication library, is also presented with theoretical estimates on speedups and efficiency, in order to examine the parallel behavior of these methods using normalized explicit approximate inverses as the suitable preconditioner. Collective communications have been utilized at the synchronization points and nonblocking communications have been used, where the exchanging of messages can be overlapped with computations, where applicable. Application of the methods on a two dimensional boundary value problem is discussed and numerical results are given, concerning the parallel performance in terms of speedups and efficiency.

show abstract

Parallel numerical linear algebra

Cited by 157 publications

References 163 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

Parallel Arnoldi method for the construction of a Krylov subspace basis: An application in magnetohydrodynamics

On the performance of parallel normalized explicit preconditioned conjugate gradient type methods

Contact Info

Product

Resources

About