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Truncation of Small Matrix Elements Based on the Euclidean Norm for Blocked Data Structures
KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
2009 (Engelska)Ingår i: Journal of Computational Chemistry, ISSN 0192-8651, E-ISSN 1096-987X, Vol. 30, nr 6, s. 974-977Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

Methods for the removal of small symmetric matrix elements based on the Euclidean norm of the error matrix are presented in this article. In large scale Hartree-Fock and Kohn-Sham calculations it is important to be able to enforce matrix sparsity while keeping errors under control. Truncation based on some unitary-invariant norm allows for control of errors in the occupied subspace as described in (Rubensson et al. J Math Phys 49, 032103). The Euclidean norm is unitary-invariant and does not grow intrinsically with system size and is thus suitable for error control in large scale calculations. The presented truncation schemes repetitively use the Lanczos method to compute the Euclidean norms of the error matrix candidates. Ritz value convergence patterns are utilized to reduce the total number of Lanczos iterations.

Ort, förlag, år, upplaga, sidor
2009. Vol. 30, nr 6, s. 974-977
Nyckelord [en]
sparsity, linear scaling, Hartree-Fock, DFT, density functional theory, blocked data structure, Euclidean norms, Lanczos, sparse matrix, Frobenius norm, electronic-structure calculations, consistent-field theory, density-matrix, expansion methods, diagonalization, minimization, purification, search
Nationell ämneskategori
Teoretisk kemi
Identifikatorer
URN: urn:nbn:se:kth:diva-18301DOI: 10.1002/jcc.21120ISI: 000264651200015PubMedID: 18816463Scopus ID: 2-s2.0-65449174900OAI: oai:DiVA.org:kth-18301DiVA, id: diva2:336347
Anmärkning
QC 20100817Tillgänglig från: 2010-08-05 Skapad: 2010-08-05 Senast uppdaterad: 2020-03-09Bibliografiskt granskad
Ingår i avhandling
1. Matrix Algebra for Quantum Chemistry
Öppna denna publikation i ny flik eller fönster >>Matrix Algebra for Quantum Chemistry
2008 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2008. s. ix, 49
Serie
Trita-BIO-Report, ISSN 1654-2312 ; 2008:23
Nyckelord
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
Nationell ämneskategori
Teoretisk kemi
Identifikatorer
urn:nbn:se:kth:diva-9447 (URN)978-91-7415-160-2 (ISBN)
Disputation
2008-11-27, FB52, Roslagstullsbacken 21, AlbaNova, 13:15 (Engelska)
Opponent
Handledare
Anmärkning
QC 20100908Tillgänglig från: 2008-11-06 Skapad: 2008-11-04 Senast uppdaterad: 2010-09-08Bibliografiskt granskad
2. Quantum Chemistry for Large Systems
Öppna denna publikation i ny flik eller fönster >>Quantum Chemistry for Large Systems
2007 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

This thesis deals with quantum chemistry methods for large systems. In particular, the thesis focuses on the efficient construction of the Coulomb and exchange matrices which are important parts of the Fock matrix in Hartree-Fock calculations. Density matrix purification, which is a method used to construct the density matrix for a given Fock matrix, is also discussed.

The methods described are not only applicable in the Hartree-Fock case, but also in Kohn-Sham Density Functional Theory calculations, where the Coulomb and exchange matrices are parts of the Kohn-Sham matrix. Screening techniques for reducing the computational complexity of both Coulomb and exchange computations are discussed, including the fast multipole method, used for efficient computation of the Coulomb matrix.

The thesis also discusses how sparsity in the matrices occurring in Hartree-Fock and Kohn-Sham Density Functional Theory calculations can be used to achieve more efficient storage of matrices as well as more efficient operations on them.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2007. s. xi, 56
Serie
Trita-BIO-Report, ISSN 1654-2312 ; 2007:13
Nyckelord
quantum chemistry, fast multipole method, density matrix purification, sparse matrices
Nationell ämneskategori
Teoretisk kemi
Identifikatorer
urn:nbn:se:kth:diva-4561 (URN)978-91-7178-797-2 (ISBN)
Disputation
2007-12-12, FA32, Albanova, Roslagstullsbacken 21, 106 91 Stockholm, 13:00
Opponent
Handledare
Anmärkning
QC 20100817Tillgänglig från: 2007-12-04 Skapad: 2007-12-04 Senast uppdaterad: 2010-08-17Bibliografiskt granskad

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