Iterative diagonalization in augmented plane wave based methods in electronic structure calculations

Blaha, Peter; Hofstätter, Harald; Koch, Othmar; Laskowski, Robert; Schwarz, K.

doi:10.1016/j.jcp.2009.09.036

Cited by 24 publications

(37 citation statements)

References 23 publications

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“…They range from robust, standard algebraic solutions (e.g., in the (Sca)LAPACK library [7]) via iterative strategies of many kinds (e.g., Refs. [1,2,8,9,10,11] and many others; best suited if only a small fraction of the eigenvalues and eigenvectors of a large matrix are needed), shape constraints on the eigenfunctions, [12], contour integration based approaches, [13] and many others, all the way to O(N ) strategies (e.g., Refs. [14,15,16] and many others) which attempt to circumvent the solution of an eigenvalue problem entirely.…”

Section: Introductionmentioning

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

244

261

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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: Introductionmentioning

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

244

261

View full text Add to dashboard Cite

show abstract

“…Thus, this example illustrates that in fact the error may increase for certain pathological problem data and hence our estimate in Theorem 4 is sharp also in such a situation. In practical computations, see for example [2], such a behavior has never been encountered, however. We set n = 5, m = 2, i.e., the dimensions as small as possible (because for m = 1 Theorem 4, eq.…”

Section: Examplementioning

confidence: 99%

“…We would like to thank P. Blaha, R. Laskowski and K. Schwarz from the Theoretical Chemistry group of the Institute for Materials Science at Vienna University of Technology for valuable discussions on the method analyzed in this paper and the possibility to realize and test the algorithm in the WIEN2k code [2].…”

Section: Acknowledgementsmentioning

confidence: 99%

“…This suggests the use of an iterative subspace method for the approximation of the eigenvalues and eigenvectors, which only computes a fraction of the eigen system which is of practical relevance. The algorithm discussed here was implemented in the WIEN2k code [4] and found to outperform previous solution methods in both computational effort and parallelization, see [2].…”

Section: Introductionmentioning

confidence: 99%

“…However, application of the subspace expansion from the Jacobi-Davidson method seems prohibitively expensive, hence we propose a simplification which uses an approximate computation of the subspace expansion related to an iterative solution of the associated linear system of equations. Furthermore, it was demonstrated in [2] that the method is superior to full diagonalization in both efficiency and parallelization.…”

Section: Introductionmentioning

confidence: 99%

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An approximate eigensolver for self-consistent field calculations

Hofstätter

2013

Numer Algor

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In this paper, we give a comprehensive error analysis for an approximate solution method for the generalized eigenvalue problems arising for instance in the context of electronic structure computations based on density functional theory. The solution method has been demonstrated to excel as compared to established solvers in both computational effort and scaling for parallelization. Here we estimate the improvement provided by our proposed subspace method starting from the initial approximations for instance provided in the course of the self-consistent field iteration, showing that in general the approximation quality is improved by our method to yield sufficiently accurate eigenvalues.

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Fast solution of Schrödinger’s equation using linear combinations of plane waves

Pérez‐Jordá

2017

Computers & Mathematics with Applications

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Iterative diagonalization in augmented plane wave based methods in electronic structure calculations

Cited by 24 publications

References 23 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

An approximate eigensolver for self-consistent field calculations

Fast solution of Schrödinger’s equation using linear combinations of plane waves

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