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- 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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