10,248 research outputs found
A Parallel Orbital-Updating Approach for Electronic Structure Calculations
In this paper, we propose an orbital iteration based parallel approach for
electronic structure calculations. This approach is based on our understanding
of the single-particle equations of independent particles that move in an
effective potential. With this new approach, the solution of the
single-particle equation is reduced to some solutions of independent linear
algebraic systems and a small scale algebraic problem. It is demonstrated by
our numerical experiments that this new approach is quite efficient for
full-potential calculations for a class of molecular systems.Comment: 22pages, 13 figures, 6 table
Novel linear algebraic theory and one-hundred-million-atom quantum material simulations on the K computer
The present paper gives a review of our recent progress and latest results
for novel linear-algebraic algorithms and its application to large-scale
quantum material simulations or electronic structure calculations. The
algorithms are Krylov-subspace (iterative) solvers for generalized shifted
linear equations, in the form of (zS-H)x=b,in stead of conventional generalized
eigen-value equation. The method was implemented in our order- calculation
code ELSES (http://www.elses.jp/) with modelled systems based on ab initio
calculations. The code realized one-hundred-million-atom, or 100-nm-scale,
quantum material simulations on the K computer in a high parallel efficiency
with up to all the built-in processor cores. The present paper also explains
several methodological aspects, such as use of XML files and 'novice' mode for
general users. A sparse matrix data library in our real problems
(http://www.elses.jp/matrix/) was prepared. Internal eigen-value problem is
discussed as a general need from the quantum material simulation. The present
study is a interdisciplinary one and is sometimes called
'Application-Algorithm-Architecture co-design'. The co-design will play a
crucial role in exa-scale scientific computations.Comment: 13 pages, 6 figure
Preconditioning orbital minimization method for planewave discretization
We present an efficient preconditioner for the orbital minimization method
when the Hamiltonian is discretized using planewaves (i.e., pseudospectral
method). This novel preconditioner is based on an approximate Fermi operator
projection by pole expansion, combined with the sparsifying preconditioner to
efficiently evaluate the pole expansion for a wide range of Hamiltonian
operators. Numerical results validate the performance of the new preconditioner
for the orbital minimization method, in particular, the iteration number is
reduced to and often only a few iterations are enough for convergence
Multilevel domain decomposition for electronic structure calculations
We introduce a new multilevel domain decomposition method (MDD) for
electronic structure calculations within semi-empirical and Density Functional
Theory (DFT) frameworks. This method iterates between local fine solvers and
global coarse solvers, in the spirit of domain decomposition methods.Comment: 36 pages, 23 figures, submitted to Journal of Computational Physic
Plane Wave Methods for Quantum Eigenvalue Problems of Incommensurate Systems
We propose a novel numerical algorithm for computing the electronic structure
related eigenvalue problem of incommensurate systems. Unlike the conventional
practice that approximates the system by a large commensurate supercell, our
algorithm directly discretizes the eigenvalue problem under the framework of a
plane wave method. The emerging ergodicity and the interpretation from higher
dimensions give rise to many unique features compared to what we have been
familiar with in the periodic system. The numerical results of 1D and 2D
quantum eigenvalue problems are presented to show the reliability and
efficiency of our scheme. Furthermore, the extension of our algorithm to full
Kohn-Sham density functional theory calculations are discussed.Comment: 21 page
ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers
Solving the electronic structure from a generalized or standard eigenproblem
is often the bottleneck in large scale calculations based on Kohn-Sham
density-functional theory. This problem must be addressed by essentially all
current electronic structure codes, based on similar matrix expressions, and by
high-performance computation. We here present a unified software interface,
ELSI, to access different strategies that address the Kohn-Sham eigenvalue
problem. Currently supported algorithms include the dense generalized
eigensolver library ELPA, the orbital minimization method implemented in
libOMM, and the pole expansion and selected inversion (PEXSI) approach with
lower computational complexity for semilocal density functionals. The ELSI
interface aims to simplify the implementation and optimal use of the different
strategies, by offering (a) a unified software framework designed for the
electronic structure solvers in Kohn-Sham density-functional theory; (b)
reasonable default parameters for a chosen solver; (c) automatic conversion
between input and internal working matrix formats, and in the future (d)
recommendation of the optimal solver depending on the specific problem.
Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800
basis functions) on distributed memory supercomputing architectures.Comment: 55 pages, 14 figures, 2 table
A partitioned shift-without-invert algorithm to improve parallel eigensolution efficiency in real-space electronic transport
We present an eigenspectrum partitioning scheme without inversion for the
recently described real-space electronic transport code, TRANSEC. The primary
advantage of TRANSEC is its highly parallel algorithm, which enables studying
conductance in large systems. The present scheme adds a new source of
parallelization, significantly enhancing TRANSEC's parallel scalability,
especially for systems with many electrons. In principle, partitioning could
enable super-linear parallel speedup, as we demonstrate in calculations within
TRANSEC. In practical cases, we report better than five-fold improvement in CPU
time and similar improvements in wall time, compared to previously-published
large calculations. Importantly, the suggested scheme is relatively simple to
implement. It can be useful for general large Hermitian or weakly non-Hermitian
eigenvalue problems, whenever relatively accurate inversion via direct or
iterative linear solvers is impractical
Higher-order adaptive finite-element methods for Kohn-Sham density functional theory
We present an efficient computational approach to perform real-space
electronic structure calculations using an adaptive higher-order finite-element
discretization of Kohn-Sham density-functional theory (DFT). To this end, we
develop an a-priori mesh adaption technique to construct a close to optimal
finite-element discretization of the problem. We further propose an efficient
solution strategy for solving the discrete eigenvalue problem by using spectral
finite-elements in conjunction with Gauss-Lobatto quadrature, and a Chebyshev
acceleration technique for computing the occupied eigenspace. The proposed
approach has been observed to provide a staggering 100-200 fold computational
advantage over the solution of a generalized eigenvalue problem. Using the
proposed solution procedure, we investigate the computational efficiency
afforded by higher-order finite-element discretization of the Kohn-Sham DFT
problem. Our studies suggest that staggering computational savings of the order
of 1000 fold relative to linear finite-elements can be realized, for both
all-electron and local pseudopotential calculations. On all the benchmark
systems studied, we observe diminishing returns in computational savings beyond
the sixth-order for accuracies commensurate with chemical accuracy. A
comparative study of the computational efficiency of the proposed higher-order
finite-element discretizations suggests that the performance of finite-element
basis is competing with the plane-wave discretization for non-periodic local
pseudopotential calculations, and compares to the Gaussian basis for
all-electron calculations within an order of magnitude. Further, we demonstrate
the capability of the proposed approach to compute the electronic structure of
a metallic system containing 1688 atoms using modest computational resources,
and good scalability of the present implementation up to 192 processors.Comment: arXiv admin note: text overlap with arXiv:1110.128
Non-linear eigensolver-based alternative to traditional SCF methods
The self-consistent procedure in electronic structure calculations is
revisited using a highly efficient and robust algorithm for solving the
non-linear eigenvector problem i.e. H({{\psi}}){\psi} = E{\psi}. This new
scheme is derived from a generalization of the FEAST eigenvalue algorithm to
account for the non-linearity of the Hamiltonian with the occupied
eigenvectors. Using a series of numerical examples and the DFT-Kohn/Sham model,
it will be shown that our approach can outperform the traditional SCF
mixing-scheme techniques by providing a higher converge rate, convergence to
the correct solution regardless of the choice of the initial guess, and a
significant reduction of the eigenvalue solve time in simulations
Multigrid Methods in Electronic Structure Calculations
We describe a set of techniques for performing large scale ab initio
calculations using multigrid accelerations and a real-space grid as a basis.
The multigrid methods provide effective convergence acceleration and
preconditioning on all length scales, thereby permitting efficient calculations
for ill-conditioned systems with long length scales or high energy cut-offs. We
discuss specific implementations of multigrid and real-space algorithms for
electronic structure calculations, including an efficient multigrid-accelerated
solver for Kohn-Sham equations, compact yet accurate discretization schemes for
the Kohn-Sham and Poisson equations, optimized pseudo\-potentials for
real-space calculations, efficacious computation of ionic forces, and a
complex-wavefunction implementation for arbitrary sampling of the Brillioun
zone. A particular strength of a real-space multigrid approach is its ready
adaptability to massively parallel computer architectures, and we present an
implementation for the Cray-T3D with essentially linear scaling of the
execution time with the number of processors. The method has been applied to a
variety of periodic and non-periodic systems, including disordered Si, a N
impurity in diamond, AlN in the wurtzite structure, and bulk Al. The high
accuracy of the atomic forces allows for large step molecular dynamics; e.g.,
in a 1 ps simulation of Si at 1100 K with an ionic step of 80 a.u., the total
energy was conserved within 27 microeV per atom.Comment: RevTeX preprint format, 41 pages, 4 postscript figure
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