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Localized inverse factorization
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
2018 (English)In: Computing Research Repository, no 1812.04919Article in journal (Other academic) Submitted
Place, publisher, year, edition, pages
2018. no 1812.04919
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-381327OAI: oai:DiVA.org:uu-381327DiVA, id: diva2:1302978
Projects
eSSENCEAvailable from: 2018-12-12 Created: 2019-04-08 Last updated: 2019-09-20Bibliographically approved
In thesis
1. Inverse factorization in electronic structure theory: Analysis and parallelization
Open this publication in new window or tab >>Inverse factorization in electronic structure theory: Analysis and parallelization
2019 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This licentiate thesis is a part of an effort to run large electronic structure calculations in modern computational environments with distributed memory. The ultimate goal is to model materials consisting of millions of atoms at the level of quantum mechanics. In particular, the thesis focuses on different aspects of a computational problem of inverse factorization of Hermitian positive definite matrices. The considered aspects are numerical properties of the algorithms and parallelization. Not only is an efficient and scalable computation of inverse factors necessary in order to be able to run large scale electronic computations based on the Hartree–Fock or Kohn–Sham approaches with the self-consistent field procedure, but it can be applied more generally for preconditioner construction.

Parallelization of algorithms with unknown load and data distributions requires a paradigm shift in programming. In this thesis we also discuss a few parallel programming models with focus on task-based models, and, more specifically, the Chunks and Tasks model.

Place, publisher, year, edition, pages
Uppsala University, 2019
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2019-002
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-381333 (URN)
Supervisors
Projects
eSSENCE
Available from: 2019-05-07 Created: 2019-04-08 Last updated: 2019-05-22Bibliographically approved
2. Efficient Density Matrix Methods for Large Scale Electronic Structure Calculations
Open this publication in new window or tab >>Efficient Density Matrix Methods for Large Scale Electronic Structure Calculations
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Efficient and accurate methods for computing the density matrix are necessary to be able to perform large scale electronic structure calculations. For sufficiently sparse matrices, the computational cost of recursive polynomial expansions to construct the density matrix scales linearly with increasing system size. In this work, parameterless stopping criteria for recursive polynomial expansions are developed. The proposed stopping criteria automatically adapt to a change in the requested accuracy, perform at almost no additional cost and do not require any user-defined tolerances.

Compared to the traditional diagonalization approach, in linear scaling methods molecular orbitals are not readily available. In this work, the interior eigenvalue problem for the Fock/Kohn-Sham matrix is coupled to the recursive polynomial expansions. The idea is to view the polynomial, obtained in the recursive expansion, as an eigenvalue filter, giving large separation between eigenvalues of interest. An efficient method for computation of homo and lumo eigenvectors is developed. Moreover, a method for computation of multiple eigenvectors around the homo-lumo gap is implemented and evaluated.

An original method for inverse factorization of Hermitian positive definite matrices is developed in this work. Novel theoretical tools for analysis of the decay properties of matrix element magnitude in electronic structure calculations are proposed. Of particular interest is an inverse factor of the basis set overlap matrix required for the density matrix construction. It is shown that the proposed inverse factorization algorithm drastically reduces the communication cost compared to state-of-the-art methods.

To perform large scale numerical tests, most of the proposed methods are implemented in the quantum chemistry program Ergo, also presented in this thesis. The recursive polynomial expansion in Ergo is parallelized using the Chunks and Tasks matrix library. It is shown that the communication cost per process of the recursive polynomial expansion implementation tends to a constant in a weak scaling setting.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2019. p. 51
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1858
Keywords
electronic structure; linear scaling; density matrix; density functional theory; Hartree-Fock; recursive polynomial expansion; density matrix purification; eigenvectors; molecular orbitals; stopping criteria; inverse factorization; matrix sparsity; parallelization
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-393387 (URN)978-91-513-0756-5 (ISBN)
Public defence
2019-11-08, 2446, ITC (Informationsteknologiskt centrum), Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2019-10-16 Created: 2019-09-20 Last updated: 2019-11-12

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https://arxiv.org/abs/1812.04919

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Rubensson, Emanuel H.Kruchinina, AnastasiaRudberg, Elias

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