arXiv subjects

Elias Rudberg

Publications and source records attributed to Elias Rudberg.

11 recordsLinked to original sources

arXiv2024

Ticket-based multi-strand method for increased efficiency in proof-of-work based blockchains

This paper outlines a method aiming to increase the efficiency of proof-of-work based blockchains using a ticket-based approach. To avoid the limitation of serially adding one block at a time to a blockchain, multiple semi-independent chains are used such that several valid blocks can be added in parallel, when they are added to separate chains. Blocks are added to different chains, the chain index being determined by a ``ticket'' that the miner must produce before creating a new block. This allows increasing the transaction rate by several orders of magnitude while the system is still fully decentralized and permissionless, and maintaining security in the sense that a successful attack would require the attacker to control a significant portion of the whole network.

cs.DC↗

arXiv2020

The Chunks and Tasks Matrix Library 2.0

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.

cs.DC↗

arXiv2019

Efficient computation of the density matrix with error control on distributed computer systems

The recursive polynomial expansion for construction of a density matrix approximation with rigorous error control [J. Chem. Phys. 128, 074106 (2008)] is implemented in the quantum chemistry program Ergo [SoftwareX 7, 107 (2018)] using the Chunks and Tasks matrix library [Parallel Comput. 57, 87 (2016)]. The expansion is based on second-order polynomials and accelerated by the scale-and-fold technique [J. Chem. Theory Comput. 7, 1233 (2011)]. We evaluate the performance of the implementation by computing the density matrix from the Fock matrix in the large-scale self-consistent field calculations. We demonstrate that the amount of communicated data per worker process tends to a constant with increasing system size and number of computer nodes such that the amount of work per worker process is fixed.

physics.comp-ph↗

arXiv2019

Localized inverse factorization

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 [J. Chem. Phys., 128 (2008), 104105] but makes use of localized operations only. At each level of recursion, the problem is cut into two subproblems and their solutions are combined using iterative refinement [Phys. Rev. B, 70 (2004), 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 proportional to the cut size, defined by the binary partition of the index set. This means that for cut sizes increasing as $o(n)$ with system size $n$ the cost of combining the two subproblems is negligible compared to the overall cost for sufficiently large systems. 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. The localization properties of our algorithms are demonstrated 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 [SoftwareX, 7 (2018), 107].

math.NA↗

arXiv2019

Parallelization and scalability analysis of inverse factorization using the Chunks and Tasks programming model

We present three methods for distributed memory parallel inverse factorization of block-sparse Hermitian positive definite matrices. The three methods are a recursive variant of the AINV inverse Cholesky algorithm, iterative refinement, and localized inverse factorization, respectively. All three methods are implemented using the Chunks and Tasks programming model, building on the distributed sparse quad-tree matrix representation and parallel matrix-matrix multiplication in the publicly available Chunks and Tasks Matrix Library (CHTML). Although the algorithms are generally applicable, this work was mainly motivated by the need for efficient and scalable inverse factorization of the basis set overlap matrix in large scale electronic structure calculations. We perform various computational tests on overlap matrices for quasi-linear Glutamic Acid-Alanine molecules and three-dimensional water clusters discretized using the standard Gaussian basis set STO-3G with up to more than 10 million basis functions. We show that for such matrices the computational cost increases only linearly with system size for all the three methods. We show both theoretically and in numerical experiments that the methods based on iterative refinement and localized inverse factorization outperform previous parallel implementations in weak scaling tests where the system size is increased in direct proportion to the number of processes. We show also that compared to the method based on pure iterative refinement the localized inverse factorization requires much less communication.

math.NA↗

arXiv2017

On-the-fly computation of frontal orbitals in density matrix expansions

Linear scaling density matrix methods typically do not provide individual eigenvectors and eigenvalues of the Fock/Kohn-Sham matrix, so additional work has to be performed if they are needed. Spectral transformation techniques facilitate computation of frontal (homo and lumo) molecular orbitals. In the purify-shift-and-square method the convergence of iterative eigenvalue solvers is improved by combining recursive density matrix expansion with the folded spectrum method [J. Chem. Phys. 128, 176101 (2008)]. However, the location of the shift in the folded spectrum method and the iteration of the recursive expansion selected for eigenpair computation may have a significant influence on the iterative eigenvalue solver performance and eigenvector accuracy. In this work, we make use of recent homo and lumo eigenvalue estimates [SIAM J. Sci. Comput. 36, B147 (2014)] for selecting shift and iteration such that homo and lumo orbitals can be computed in a small fraction of the total recursive expansion time and with sufficient accuracy. We illustrate our method by performing self-consistent field calculations for large scale systems.

physics.comp-ph↗

arXiv2016

Parameterless stopping criteria for recursive density matrix expansions

Parameterless stopping criteria for recursive polynomial expansions to construct the density matrix in electronic structure calculations are proposed. Based on convergence order estimation the new stopping criteria automatically and accurately detect when the calculation is dominated by numerical errors and continued iteration does not improve the result. Difficulties in selecting a stopping tolerance and appropriately balancing it in relation to parameters controlling the numerical accuracy are avoided. Thus, our parameterless stopping criteria stands in contrast to the standard approach to stop as soon as some error measure goes below a user-defined parameter or tolerance. We demonstrate that the stopping criteria work well both in dense and sparse matrix calculations and in large-scale self-consistent field calculations with the quantum chemistry program Ergo (www.ergoscf.org).

physics.comp-ph↗

arXiv2016

Locality-aware parallel block-sparse matrix-matrix multiplication using the Chunks and Tasks programming model

We present a method for parallel block-sparse matrix-matrix multiplication on distributed memory clusters. By using a quadtree matrix representation, data locality is exploited without prior information about the matrix sparsity pattern. A distributed quadtree matrix representation is straightforward to implement due to our recent development of the Chunks and Tasks programming model [Parallel Comput. 40, 328 (2014)]. The quadtree representation combined with the Chunks and Tasks model leads to favorable weak and strong scaling of the communication cost with the number of processes, as shown both theoretically and in numerical experiments. Matrices are represented by sparse quadtrees of chunk objects. The leaves in the hierarchy are block-sparse submatrices. Sparsity is dynamically detected by the matrix library and may occur at any level in the hierarchy and/or within the submatrix leaves. In case graphics processing units (GPUs) are available, both CPUs and GPUs are used for leaf-level multiplication work, thus making use of the full computing capacity of each node. The performance is evaluated for matrices with different sparsity structures, including examples from electronic structure calculations. Compared to methods that do not exploit data locality, our locality-aware approach reduces communication significantly, achieving essentially constant communication per node in weak scaling tests.

cs.DC↗

arXiv2015

Canonical density matrix perturbation theory

Density matrix perturbation theory [Niklasson and Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] is generalized to canonical (NVT) free energy ensembles in tight-binding, Hartree-Fock or Kohn-Sham density functional theory. The canonical density matrix perturbation theory can be used to calculate temperature dependent response properties from the coupled perturbed self-consistent field equations as in density functional perturbation theory. The method is well suited to take advantage of sparse matrix algebra to achieve linear scaling complexity in the computational cost as a function of system size for sufficiently large non-metallic materials and metals at high temperatures.

physics.chem-ph↗

arXiv2012

Chunks and Tasks: a programming model for parallelization of dynamic algorithms

We propose Chunks and Tasks, a parallel programming model built on abstractions for both data and work. The application programmer specifies how data and work can be split into smaller pieces, chunks and tasks, respectively. The Chunks and Tasks library maps the chunks and tasks to physical resources. In this way we seek to combine user friendliness with high performance. An application programmer can express a parallel algorithm using a few simple building blocks, defining data and work objects and their relationships. No explicit communication calls are needed; the distribution of both work and data is handled by the Chunks and Tasks library. This makes efficient implementation of complex applications that require dynamic distribution of work and data easier. At the same time, Chunks and Tasks imposes restrictions on data access and task dependencies that facilitates the development of high performance parallel back ends. We discuss the fundamental abstractions underlying the programming model, as well as performance and fault resilience considerations. We also present a pilot C++ library implementation for clusters of multicore machines and demonstrate its performance for sparse blocked matrix-matrix multiplication.

cs.DC↗

arXiv2011

Difficulties in applying pure Kohn-Sham density functional theory electronic structure methods to protein molecules

Self-consistency based Kohn-Sham density functional theory (KS-DFT) electronic structure calculations with Gaussian basis sets are reported for a set of 17 protein-like molecules with geometries obtained from the protein data bank. It is found that in many cases such calculations do not converge due to vanishing HOMO-LUMO gaps. A sequence of polyproline I helix molecules is also studied, and it is found that self-consistency calculations using pure functionals fail to converge for helices longer than six proline units. Since the computed gap is strongly correlated to the fraction of Hartree-Fock exchange, test calculations using both pure and hybrid density functionals are reported. The tested methods include the pure functionals BLYP, PBE, and LDA, as well as Hartree-Fock and the hybrid functionals BHandHLYP, B3LYP, and PBE0. The effect of including solvent molecules in the calculations is studied, and it is found that the inclusion of explicit solvent molecules around the protein fragment in many cases gives a larger gap, but that convergence problems due to vanishing gaps still occur in calculations with pure functionals. In order to achieve converged results, some modeling of the charge distribution of solvent water molecules outside the electronic structure calculation is needed. Representing solvent water molecules by a simple point charge distribution is found to give non-vanishing HOMO-LUMO gaps for the tested protein-like systems also for pure functionals.

physics.chem-ph↗