| 1. |
- Artemov, Anton G.
(författare)
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Inverse factorization in electronic structure theory : Analysis and parallelization
- 2019
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 2. |
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| 3. |
- Artemov, Anton G., 1990-
(författare)
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Parallelization of dynamic algorithms for electronic structure calculations
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The aim of electronic structure calculations is to simulate behavior of complex materials by resolving interactions between electrons and nuclei in atoms at the level of quantum mechanics. Progress in the field allows to reduce the computational complexity of the solution methods to linear so that the computational time scales proportionally to the size of the physical system. To solve large scale problems one uses parallel computers and scalable codes. Often the scalability is limited by the data distribution.This thesis focuses on a number of problems arising in electronic structure calculations, such as inverse factorization of Hermitian positive definite matrices, approximate sparse matrix multiplication, and density matrix purification methods. No assumptions are made about the data distribution, instead, it is explored dynamically.The thesis consists of an introduction and five papers. Particularly, in Paper I we present a new theoretical framework for localized matrices with exponential decay of elements. We describe a new localized method for inverse factorization of Hermitian positive definite matrices. We show that it has reduced communication costs compared to other widely used parallel methods. In Paper II we present a parallel implementation of the method within the Chunks and Tasks programming model and do a scalability analysis based on critical path length estimation.We focus on the density matrix purification technique and its core operation, sparse matrix-matrix multiplication, in Papers III and IV. We analyze the sparse approximate matrix multiplication algorithm with the proposed localization framework, add a prior truncation step, and derive the asymptotic behavior of the Frobenius norm of the error. We employ the sparse approximate multiplication algorithm in the density matrix purification process and propose a method to control the error norm by choosing the right truncation threshold value. We present a new version of the Chunks and Tasks matrix library in Paper V. The library functionality and architecture are described and discussed. The efficiency of the library is demonstrated in a few computational experiments.
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| 4. |
- Artemov, Anton G., et al.
(författare)
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Sparse approximate matrix-matrix multiplication for density matrix purification with error control
- 2021
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 438
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Tidskriftsartikel (refereegranskat)abstract
- We propose an accelerated density matrix purification scheme with error control. The method makes use of the scale-and-fold acceleration technique and screening of submatrix products in the block-sparse matrix-matrix multiplies to reduce the computational cost. An error bound and a parameter sweep are combined to select a threshold value for the screening, such that the error can be controlled. We evaluate the performance of the method in comparison to purification without acceleration and without submatrix product screening.
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| 5. |
- Finkelstein, Joshua, et al.
(författare)
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Mixed Precision Fermi-Operator Expansion on Tensor Cores from a Machine Learning Perspective
- 2021
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Ingår i: Journal of Chemical Theory and Computation. - : American Chemical Society (ACS). - 1549-9618 .- 1549-9626. ; 17:4, s. 2256-2265
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Tidskriftsartikel (refereegranskat)abstract
- We present a second-order recursive Fermi-operator expansion scheme using mixed precision floating point operations to perform electronic structure calculations using tensor core units. A performance of over 100 teraFLOPs is achieved for half-precision floating point operations on Nvidia’s A100 tensor core units. The second-order recursive Fermi-operator scheme is formulated in terms of a generalized, differentiable deep neural network structure, which solves the quantum mechanical electronic structure problem. We demonstrate how this network can be accelerated by optimizing the weight and bias values to substantially reduce the number of layers required for convergence. We also show how this machine learning approach can be used to optimize the coefficients of the recursive Fermi-operator expansion to accurately represent the fractional occupation numbers of the electronic states at finite temperatures.
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| 6. |
- Finkelstein, Joshua, et al.
(författare)
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Quantum-Based Molecular Dynamics Simulations Using Tensor Cores
- 2021
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Ingår i: Journal of Chemical Theory and Computation. - : American Chemical Society (ACS). - 1549-9618 .- 1549-9626. ; 17:10, s. 6180-6192
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Tidskriftsartikel (refereegranskat)abstract
- Tensor cores, along with tensor processing units, represent a new form of hardware acceleration specifically designed for deep neural network calculations in artificial intelligence applications. Tensor cores provide extraordinary computational speed and energy efficiency but with the caveat that they were designed for tensor contractions (matrix-matrix multiplications) using only low-precision floating-point operations. Despite this perceived limitation, we demonstrate how tensor cores can be applied with high efficiency to the challenging and numerically sensitive problem of quantum-based Born-Oppenheimer molecular dynamics, which requires highly accurate electronic structure optimizations and conservative force evaluations. The interatomic forces are calculated on-the-fly from an electronic structure that is obtained from a generalized deep neural network, where the computational structure naturally takes advantage of the exceptional processing power of the tensor cores and allows for high performance in excess of 100 Tflops on a single Nvidia A100 GPU. Stable molecular dynamics trajectories are generated using the framework of extended Lagrangian Born-Oppenheimer molecular dynamics, which combines computational efficiency with long-term stability, even when using approximate charge relaxations and force evaluations that are limited in accuracy by the numerically noisy conditions caused by the low-precision tensor core floating-point operations. A canonical ensemble simulation scheme is also presented, where the additional numerical noise in the calculated forces is absorbed into a Langevin-like dynamics.
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| 7. |
- Finkelstein, Joshua, et al.
(författare)
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Quantum Perturbation Theory Using Tensor Cores and a Deep Neural Network
- 2022
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Ingår i: Journal of Chemical Theory and Computation. - : American Chemical Society (ACS). - 1549-9618 .- 1549-9626. ; 18:7, s. 4255-4268
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Tidskriftsartikel (refereegranskat)abstract
- Time-independent quantum response calculations are performed using Tensor cores. This is achieved by mapping density matrix perturbation theory onto the computational structure of a deep neural network. The main computational cost of each deep layer is dominated by tensor contractions, i.e., dense matrix–matrix multiplications, in mixed-precision arithmetics, which achieves close to peak performance. Quantum response calculations are demonstrated and analyzed using self-consistent charge density-functional tight-binding theory as well as coupled-perturbed Hartree–Fock theory. For linear response calculations, a novel parameter-free convergence criterion is presented that is well-suited for numerically noisy low-precision floating point operations and we demonstrate a peak performance of almost 200 Tflops using the Tensor cores of two Nvidia A100 GPUs.
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| 8. |
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| 9. |
- Kruchinina, Anastasia, 1991-
(författare)
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Efficient Density Matrix Methods for Large Scale Electronic Structure Calculations
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 10. |
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| 11. |
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| 12. |
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| 13. |
- Niklasson, Anders M. N., et al.
(författare)
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Canonical density matrix perturbation theory
- 2015
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Ingår i: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics. - 1539-3755 .- 1550-2376. ; 92, s. 063301:1-8
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Tidskriftsartikel (refereegranskat)
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| 14. |
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| 15. |
- Rubensson, Emanuel H., et al.
(författare)
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A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations
- 2007
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Ingår i: Journal of Computational Chemistry. - : Wiley. - 0192-8651 .- 1096-987X. ; 28:16, s. 2531-2537
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Tidskriftsartikel (refereegranskat)abstract
- A hierarchic sparse matrix data structure for Hartree-Fock/Kohn-Sham calculations is presented. The data structure makes the implementation of matrix manipulations needed for large systems faster, easier, and more maintainable without loss of performance. Algorithms for symmetric matrix square and inverse Cholesky decomposition within the hierarchic framework are also described. The presented data structure is general; in addition to its use in HartreeFock/Kohn-Sham calculations, it may also be used in other research areas where matrices with similar properties are encountered. The applicability of the data structure to ab initio calculations is shown with help of benchmarks on water droplets and graphene nanoribbons.
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| 16. |
- Rubensson, Emanuel H.
(författare)
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A unifying framework for higher order derivatives of matrix functions
- 2024
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Ingår i: SIAM Journal on Matrix Analysis and Applications. - : Society for Industrial and Applied Mathematics. - 0895-4798 .- 1095-7162. ; 45:1, s. 504-528
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Tidskriftsartikel (refereegranskat)abstract
- We present theory for general partial derivatives of matrix functions of the form f(A(x)) where A(x) is a matrix path of several variables (x=(x1,...,xj)). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610-620] for the first order derivative, we develop a block upper triangular form for higher order partial derivatives. This block form is used to derive conditions for existence and a generalized Daleckii-Krein formula for higher order derivatives. We show that certain specializations of this formula lead to classical formulas of quantum perturbation theory. We show how our results are related to earlier results for higher order Fréchet derivatives. Block forms of complex step approximations are introduced and we show how those are related to evaluation of derivatives through the upper triangular form. These relations are illustrated with numerical examples.
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| 17. |
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| 18. |
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| 19. |
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| 20. |
- Rubensson, Emanuel H., 1979-, et al.
(författare)
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Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification
- 2008
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 128:17, s. 176101-
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Tidskriftsartikel (refereegranskat)abstract
- Density matrix purification, is in this work, used to facilitate the computation of eigenpairs around the highest occupied and the lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) in electronic structure calculations. The ability of purification to give large separation between eigenvalues close to the HOMO-LUMO gap is used to accelerate convergence of the Lanczos method. Illustrations indicate that a new eigenpair is found more often than every second Lanczos iteration when the proposed methods are used.
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| 21. |
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| 22. |
- Rubensson, Emanuel H., et al.
(författare)
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Density matrix purification with rigorous error control
- 2008
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 128:7
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Tidskriftsartikel (refereegranskat)abstract
- Density matrix purification, although being a powerful tool for linear scaling construction of the density matrix in electronic structure calculations, has been limited by uncontrolled error accumulation. In this article, a strategy for the removal of small matrix elements in density matrix purification is proposed with which the forward error can be rigorously controlled. The total forward error is separated into two parts, the error in eigenvalues and the error in the occupied invariant subspace. We use the concept of canonical angles to measure and control differences between exact and approximate occupied subspaces. We also analyze the conditioning of the density matrix construction problem and propose a method for calculation of interior eigenvalues to be used together with density matrix purification. (C) 2008 American Institute of Physics.
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| 23. |
- Rubensson, Emanuel H., et al.
(författare)
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Interior eigenvalues from density matrix expansions in quantum mechanical molecular dynamics
- 2014
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 36, s. B147-B170
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Tidskriftsartikel (refereegranskat)abstract
- An accelerated polynomial expansion scheme to construct the density matrix in quantum mechanical molecular dynamics simulations is proposed. The scheme is based on recursive density matrix expansions, e. g., [A. M. N. Niklasson, Phys. Rev. B, 66 (2002), 155115], which are accelerated by a scale-and-fold technique [E. H. Rubensson, J. Chem. Theory Comput., 7 (2011), pp. 1233-1236]. The acceleration scheme requires interior eigenvalue estimates, which may be expensive and cumbersome to come by. Here we show how such eigenvalue estimates can be extracted from the recursive expansion by a simple and robust procedure at a negligible computational cost. Our method is illustrated with density functional tight-binding Born-Oppenheimer molecular dynamics simulations, where the computational effort is dominated by the density matrix construction. In our analysis we identify two different phases of the recursive polynomial expansion, the conditioning and purification phases, and we show that the acceleration represents an improvement of the conditioning phase, which typically gives a significant reduction of the computational cost.
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| 24. |
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| 25. |
- Rubensson, Emanuel H., et al.
(författare)
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Localized inverse factorization
- 2021
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Ingår i: IMA Journal of Numerical Analysis. - : Oxford University Press (OUP). - 0272-4979 .- 1464-3642. ; 41:1, s. 729-763
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Tidskriftsartikel (refereegranskat)abstract
- We propose a localized divide and conquer algorithm for inverse factorization S-1 = ZZ* of Hermitian positive definite matrices S with localized structure, e.g. exponential decay with respect to some given distance function on the index set of S. The algorithm is a reformulation of recursive inverse factorization (Rubensson et al. (2008) Recursive inverse factorization. J. Chem. Phys., 128, 104105) but makes use of localized operations only. At each level of the recursion, the problem is cut into two subproblems and their solutions are combined using iterative refinement (Niklasson (2004) Iterative refinement method for the approximate factorization of a matrix inverse. Phys. Rev. B, 70, 193102) to give a solution to the original problem. The two subproblems can be solved in parallel without any communication and, using the localized formulation, the cost of combining their results is negligible compared to the overall cost for sufficiently large systems and appropriate partitions of the problem. We also present an alternative derivation of iterative refinement based on a sign matrix formulation, analyze the stability and propose a parameterless stopping criterion. We present bounds for the initial factorization error and the number of iterations in terms of the condition number of S when the starting guess is given by the solution of the two subproblems in the binary recursion. These bounds are used in theoretical results for the decay properties of the involved matrices. We demonstrate the localization properties of our algorithm for matrices corresponding to nearest neighbor overlap on one-, two- and three-dimensional lattices, as well as basis set overlap matrices generated using the Hartree-Fock and Kohn-Sham density functional theory electronic structure program Ergo (Rudberg et al. (2018) Ergo: an open-source program for linear-scaling electronic structure. SoftwareX, 7, 107). We evaluate the parallel performance of our implementation based on the chunks and tasks programming model, showing that the proposed localization of the algorithm results in a dramatic reduction of communication costs.
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| 26. |
- Rubensson, Emanuel H., 1979-
(författare)
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Matrix Algebra for Quantum Chemistry
- 2008
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 27. |
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| 28. |
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| 29. |
- Rubensson, Emanuel H., 1979-, et al.
(författare)
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Recursive inverse factorization
- 2008
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 128:10, s. 104105-
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 30. |
- Rubensson, Emanuel H., et al.
(författare)
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Rotations of occupied invariant subspaces in self-consistent field calculations
- 2008
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 49:3, s. 032103-
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Tidskriftsartikel (refereegranskat)abstract
- In this article, the self-consistent field (SCF) procedure as used in Hartree-Fock and Kohn-Sham calculations is viewed as a sequence of rotations of the so-called occupied invariant subspace of the potential and density matrices. Computational approximations are characterized as erroneous rotations of this subspace. Differences between subspaces are measured and controlled by the canonical angles between them. With this approach, a first step is taken toward a method where errors from computational approximations are rigorously controlled and threshold values are directly related to the accuracy of the current trial density, thus eliminating the use of ad hoc threshold values. Then, the use of computational resources can be kept down as much as possible without impairment of the SCF convergence. (C) 2008 American Institute of Physics.
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| 31. |
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| 32. |
- Rubensson, Emanuel H., et al.
(författare)
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The Chunks and Tasks Matrix Library
- 2022
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Ingår i: SoftwareX. - : Elsevier. - 2352-7110. ; 19
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Tidskriftsartikel (refereegranskat)abstract
- We present a C++ header-only parallel sparse matrix library, based on sparse quadtree representation of matrices using the Chunks and Tasks programming model. The library implements a number of sparse matrix algorithms for distributed memory parallelization that are able to dynamically exploit data locality to avoid movement of data. This is demonstrated for the example of block-sparse matrix-matrix multiplication applied to three sequences of matrices with different nonzero structure, using the CHT-MPI 2.0 runtime library implementation of the Chunks and Tasks model. The runtime library succeeds to dynamically load balance the calculation regardless of the sparsity structure.
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| 33. |
- Rubensson, Emanuel H., et al.
(författare)
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Truncation of Small Matrix Elements Based on the Euclidean Norm for Blocked Data Structures
- 2009
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Ingår i: Journal of Computational Chemistry. - : Wiley. - 0192-8651 .- 1096-987X. ; 30:6, s. 974-977
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 34. |
- Rudberg, Elias, et al.
(författare)
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Assessment of density matrix methods for electronic structure calculations
- 2010
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Purification and minimization methods for computation of the one-particle density matrix are compared. This is done by considering the work needed by each method to achieve a given accuracy in terms of the difference to the exact solution. Simulations employing orthogonal as well as non-orthogonal versions of the methods are performed using both element magnitude and cutoff radius based truncation approaches. The results indicate that purification is considerably more efficient than the studied minimization methods even when a good starting guess for minimization is available. The computational cost of the studied minimization methods is observed to be significantly more sensitive to small band gaps than purification. An O(sqrt(1/xi)) dependence on the band gap xi is observed for minimization which can be compared to the O(ln{(1/xi)}) dependence for purification. Minimization is found to perform at its best at 50% occupancy. Error control and stopping criteria are also discussed.
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| 35. |
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| 36. |
- Rudberg, Elias, et al.
(författare)
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Automatic Selection of Integral Thresholds by Extrapolation in Coulomb and Exchange Matrix Constructions
- 2009
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Ingår i: Journal of Chemical Theory and Computation. - : American Chemical Society (ACS). - 1549-9618 .- 1549-9626. ; 5:1, s. 80-85
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Tidskriftsartikel (refereegranskat)abstract
- We present a method to compute Coulomb and exchange matrices with predetermined accuracy as measured by a matrix norm. The computation of these matrices is fundamental in Hartree-Fock and Kohn-Sham electronic structure calculations. We show numerically that, when modern algorithms for Coulomb and exchange matrix evaluation are applied, the Euclidean norm of the error matrix e is related to the threshold value tau as epsilon C tau(alpha). The presented extrapolation method automatically selects the integral thresholds so that the Euclidean norm of the error matrix is at the requested accuracy. This approach is demonstrated for a variety of systems, including protein-like systems, water clusters, and graphene sheets. The proposed method represents an important step toward complete error control throughout the self-consistent field calculation as described in [J. Math. Phys. 2008, 49, 032103].
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| 37. |
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| 38. |
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| 39. |
- Rudberg, Elias, et al.
(författare)
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Hartree-Fock calculations with linearly scaling memory usage
- 2008
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Ingår i: Journal of Chemical Physics. - : AIP Publishing. - 0021-9606 .- 1089-7690. ; 128:18
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Tidskriftsartikel (refereegranskat)abstract
- We present an implementation of a set of algorithms for performing Hartree-Fock calculations with resource requirements in terms of both time and memory directly proportional to the system size. In particular, a way of directly computing the Hartree-Fock exchange matrix in sparse form is described which gives only small addressing overhead. Linear scaling in both time and memory is demonstrated in benchmark calculations for system sizes up to 11 650 atoms and 67 204 Gaussian basis functions on a single computer with 32 Gbytes of memory. The sparsity of overlap, Fock, and density matrices as well as band gaps are also shown for a wide range of system sizes, for both linear and three-dimensional systems. (C) 2008 American Institute of Physics.
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| 40. |
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| 41. |
- Upadhyaya, Parikshit, et al.
(författare)
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A density matrix approach to the convergence of the self-consistent field iteration
- 2021
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Ingår i: Numerical Algebra, Control and Optimization. - : American Institute of Mathematical Sciences (AIMS). - 2155-3289 .- 2155-3297. ; 11:1, s. 99-115
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we present a local convergence analysis of the self-consistent field (SCF) iteration using the density matrix as the state of a fixed-point iteration. Conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map. The relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in terms of higher gaps. This gives more information regarding how the gaps between eigenvalues of the problem affect the convergence, and hence these bounds are more insightful on the convergence behaviour than standard convergence results. We also provide a detailed analysis to describe the difference between the bounds and the exact convergence factor for an illustrative example. Finally we present numerical examples and compare the exact value of the convergence factor with the observed behaviour of SCF, along with our new bounds and the characterization using the higher gaps. We provide heuristic convergence factor estimates in situations where the bounds fail to well capture the convergence.
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