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Systematic sparse matrix error control for linear scaling electronic structure calculations
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
2005 (English)In: Journal of Computational Chemistry, ISSN 0192-8651, E-ISSN 1096-987X, Vol. 26, no 15, 1628-1637 p.Article in journal (Refereed) Published
Abstract [en]

 Efficient truncation criteria used in multiatom blocked sparse matrix operations for ab initio calculations are proposed. As system size increases, so does the need to stay on top of errors and still achieve high performance. A variant of a blocked sparse matrix algebra to achieve strict error control with good performance is proposed. The presented idea is that the condition to drop a certain submatrix should depend not only on the magnitude of that particular submatrix, but also on which other submatrices that are dropped. The decision to remove a certain submatrix is based on the contribution the removal would cause to the error in the chosen norm. We study the effect of an accumulated truncation error in iterative algorithms like trace correcting density matrix purification. One way to reduce the initial exponential growth of this error is presented. The presented error control for a sparse blocked matrix toolbox allows for achieving optimal performance by performing only necessary operations needed to maintain the requested level of accuracy.

Place, publisher, year, edition, pages
2005. Vol. 26, no 15, 1628-1637 p.
Keyword [en]
matrix error control, linear scaling, density purification
National Category
Theoretical Chemistry
Identifiers
URN: urn:nbn:se:kth:diva-6544DOI: 10.1002/jcc.20315ISI: 000232570300008Scopus ID: 2-s2.0-27844477884OAI: oai:DiVA.org:kth-6544DiVA: diva2:11286
Note
QC 20101123Available from: 2006-12-11 Created: 2006-12-11 Last updated: 2010-11-23Bibliographically approved
In thesis
1. Sparse Matrices in Self-Consistent Field Methods
Open this publication in new window or tab >>Sparse Matrices in Self-Consistent Field Methods
2006 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis is part of an effort to enable large-scale Hartree-Fock/Kohn-Sham (HF/KS) calculations. The objective is to model molecules and materials containing thousands of atoms at the quantum mechanical level. HF/KS calculations are usually performed with the Self-Consistent Field (SCF) method. This method involves two computationally intensive steps. These steps are the construction of the Fock/Kohn-Sham potential matrix from a given electron density and the subsequent update of the electron density usually represented by the so-called density matrix. In this thesis the focus lies on the representation of potentials and electron density and on the density matrix construction step in the SCF method. Traditionally a diagonalization has been used for the construction of the density matrix. This diagonalization method is, however, not appropriate for large systems since the time complexity for this operation is σ(n3). Three types of alternative methods are described in this thesis; energy minimization, Chebyshev expansion, and density matrix purification. The efficiency of these methods relies on fast matrix-matrix multiplication. Since the occurring matrices become sparse when the separation between atoms exceeds some value, the matrix-matrix multiplication can be performed with complexity σ(n).

A hierarchic sparse matrix data structure is proposed for the storage and manipulation of matrices. This data structure allows for easy development and implementation of algebraic matrix operations, particularly needed for the density matrix construction, but also for other parts of the SCF calculation. The thesis addresses also truncation of small elements to enforce sparsity, permutation and blocking of matrices, and furthermore calculation of the HOMO-LUMO gap and a few surrounding eigenpairs when density matrix purification is used instead of the traditional diagonalization method.

Place, publisher, year, edition, pages
Stockholm: Bioteknologi, 2006. x, 38 p.
Keyword
sparse matrix, self-consistent field, Hartree-Fock, Density Functional Theory, Density Matrix Purification
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-4219 (URN)978-91-7178-534-3 (ISBN)978-91-7178-534-5 (ISBN)
Presentation
2006-12-15, FD41, AlbaNova, Roslagstullsbacken 21, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20101123Available from: 2006-12-11 Created: 2006-12-11 Last updated: 2010-11-23Bibliographically approved

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