Rapid filtration algorithm to construct a minimal basis on the fly from a primitive Gaussian basis

Rayson, Mark

doi:10.1016/j.cpc.2010.02.012

Cited by 45 publications

(47 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Solving or even circumventing the solution of (1) is thus an active research field, with many new and original contributions even in the most recent literature [6,7,9,[11][12][13][14][15][16][17][18][19]. Among the strategies pursued in electronic structure theory, one finds:…”

Section: Introductionmentioning

confidence: 99%

“…Other choices, e.g., Slater-type or numerically tabulated atom-centered orbitals (NAOs) (see [8] for references and discussion), can yield essentially converged accuracy with even more compact basis sets that still remain generically transferable. Finally, a recent localization-based filtering approach [7,19] contracts a large, generic basis set into a system-dependent minimal basis (n = k), prior to solving (1). The O(N 3 ) bottleneck is then reduced to the minimum…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

et al. 2011

View full text Add to dashboard Cite

a b s t r a c tThe computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute the eigenpairs of the tridiagonal matrix, and transform the eigenvectors back. To better utilize memory hierarchies, the reduction may be effected in two stages: full to banded, and banded to tridiagonal. Then the back transformation of the eigenvectors also involves two stages. For large problems, the eigensystem calculations can be the computational bottleneck, in particular with large numbers of processors. In this paper we discuss variants of the tridiagonal-to-banded back transformation, improving the parallel efficiency for large numbers of processors as well as the per-processor utilization. We also modify the divide-and-conquer algorithm for symmetric tridiagonal matrices such that it can compute a subset of the eigenpairs at reduced cost. The effectiveness of our modifications is demonstrated with numerical experiments.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

et al. 2011

View full text Add to dashboard Cite

show abstract

“…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

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

“…Recently, adaptive basis schemes that do not require the global Hamiltonian or density matrix have been presented. The localized filter diagonalization (LFD) method builds an adaptive basis on-the-fly by contracting the atomic Gaussian functions within a local region, with contraction coefficients determined by diagonalizing a block of the Hamiltonian matrix corresponding to that region 98,99 . This algorithm has also been used to construct multisite local support functions 100 , and the general philosophy has been extended by Lin et al 101 , including another model with more rigorous optimization 102 .…”

Section: Kohn-sham Density Functional Theorymentioning

confidence: 99%

Approaching the basis set limit for DFT calculations using an environment-adapted minimal basis with perturbation theory: Formulation, proof of concept, and a pilot implementation

Mao

Horn

Mardirossian

et al. 2016

The Journal of Chemical Physics

View full text Add to dashboard Cite

Recently developed density functionals have good accuracy for both thermochemistry (TC) and non-covalent interactions (NC) if very large basis sets are used. To approach the basis set limit with potentially lower costs, a new SCF scheme with minimal adaptive basis (MAB) functions variationally optimized on each atomic site is presented, and the desired accuracy can be obtained by applying a perturbative correction (PC). Compared to exact SCF results, using this MAB-SCF (PC) approach with the same target basis set produces < 0.15 kcal/mol RMS errors for most of the tested TC datasets, and < 0.1 kcal/mol for most of the NCs. The performance of density functionals near the basis set limit can be well reproduced with slight discrepancies. With further improvement to its implementation, MAB-SCF (PC) could be a promising substitute for its conventional counterpart to viably approach the basis set limit of modern density functionals.

show abstract

Rapid filtration algorithm to construct a minimal basis on the fly from a primitive Gaussian basis

Cited by 45 publications

References 15 publications

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

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

Approaching the basis set limit for DFT calculations using an environment-adapted minimal basis with perturbation theory: Formulation, proof of concept, and a pilot implementation

Contact Info

Product

Resources

About