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Recursive inverse factorization
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
Theoretical Division, Los Alamos National Laboratory.
Instituto de Física, Universidad Austral de Chile.
KTH, School of Industrial Engineering and Management (ITM), Materials Science and Engineering, Applied Material Physics.
2008 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, no 10, 104105- p.Article in journal (Refereed) Published
Abstract [en]

A recursive algorithm for the inverse factorization S−1=ZZ* of Hermitian positive definite matrices S is proposed. The inverse factorization is based on iterative refinement [A.M.N. Niklasson, Phys. Rev. B 70, 193102 (2004)] combined with a recursive decomposition of S. As the computational kernel is matrix-matrix multiplication, the algorithm can be parallelized and the computational effort increases linearly with system size for systems with sufficiently sparse matrices. Recent advances in network theory are used to find appropriate recursive decompositions. We show that optimization of the so-called network modularity results in an improved partitioning compared to other approaches. In particular, when the recursive inverse factorization is applied to overlap matrices of irregularly structured three-dimensional molecules.

Place, publisher, year, edition, pages
2008. Vol. 128, no 10, 104105- p.
Keyword [en]
Hermitian matrices, matrix decomposition, recursion method, sparse matrices, congruence transformation, inverse factorization, iterative refinement
National Category
Theoretical Chemistry
URN: urn:nbn:se:kth:diva-9446DOI: 10.1063/1.2884921ISI: 000254025300007ScopusID: 2-s2.0-40849134177OAI: diva2:114025
QC 20100908Available from: 2008-11-04 Created: 2008-11-04 Last updated: 2010-09-08Bibliographically approved
In thesis
1. Matrix Algebra for Quantum Chemistry
Open this publication in new window or tab >>Matrix Algebra for Quantum Chemistry
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis concerns methods of reduced complexity for electronic structure calculations.  When quantum chemistry methods are applied to large systems, it is important to optimally use computer resources and only store data and perform operations that contribute to the overall accuracy. At the same time, precarious approximations could jeopardize the reliability of the whole calculation.  In this thesis, the self-consistent field method is seen as a sequence of rotations of the occupied subspace. Errors coming from computational approximations are characterized as erroneous rotations of this subspace. This viewpoint is optimal in the sense that the occupied subspace uniquely defines the electron density. Errors should be measured by their impact on the overall accuracy instead of by their constituent parts. With this point of view, a mathematical framework for control of errors in Hartree-Fock/Kohn-Sham calculations is proposed.  A unifying framework is of particular importance when computational approximations are introduced to efficiently handle large systems.

An important operation in Hartree-Fock/Kohn-Sham calculations is the calculation of the density matrix for a given Fock/Kohn-Sham matrix. In this thesis, density matrix purification is used to compute the density matrix with time and memory usage increasing only linearly with system size. The forward error of purification is analyzed and schemes to control the forward error are proposed. The presented purification methods are coupled with effective methods to compute interior eigenvalues of the Fock/Kohn-Sham matrix also proposed in this thesis.New methods for inverse factorizations of Hermitian positive definite matrices that can be used for congruence transformations of the Fock/Kohn-Sham and density matrices are suggested as well.

Most of the methods above have been implemented in the Ergo quantum chemistry program. This program uses a hierarchic sparse matrix library, also presented in this thesis, which is parallelized for shared memory computer architectures. It is demonstrated that the Ergo program is able to perform linear scaling Hartree-Fock calculations.

Place, publisher, year, edition, pages
Stockholm: KTH, 2008. ix, 49 p.
Trita-BIO-Report, ISSN 1654-2312 ; 2008:23
linear scaling, reduced complexity, electronic structure, density functional theory, Hartree-Fock, density matrix purification, congruence transformation, inverse factorization, eigenvalues, eigenvectors, numerical linear algebra, occupied subspace, canonical angles, invariant subspace
National Category
Theoretical Chemistry
urn:nbn:se:kth:diva-9447 (URN)978-91-7415-160-2 (ISBN)
Public defence
2008-11-27, FB52, Roslagstullsbacken 21, AlbaNova, 13:15 (English)
QC 20100908Available from: 2008-11-06 Created: 2008-11-04 Last updated: 2010-09-08Bibliographically approved

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