5,757 research outputs found
Symbolic-numeric interface: A review
A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach
Fast, accurate, and transferable many-body interatomic potentials by symbolic regression
The length and time scales of atomistic simulations are limited by the
computational cost of the methods used to predict material properties. In
recent years there has been great progress in the use of machine learning
algorithms to develop fast and accurate interatomic potential models, but it
remains a challenge to develop models that generalize well and are fast enough
to be used at extreme time and length scales. To address this challenge, we
have developed a machine learning algorithm based on symbolic regression in the
form of genetic programming that is capable of discovering accurate,
computationally efficient manybody potential models. The key to our approach is
to explore a hypothesis space of models based on fundamental physical
principles and select models within this hypothesis space based on their
accuracy, speed, and simplicity. The focus on simplicity reduces the risk of
overfitting the training data and increases the chances of discovering a model
that generalizes well. Our algorithm was validated by rediscovering an exact
Lennard-Jones potential and a Sutton Chen embedded atom method potential from
training data generated using these models. By using training data generated
from density functional theory calculations, we found potential models for
elemental copper that are simple, as fast as embedded atom models, and capable
of accurately predicting properties outside of their training set. Our approach
requires relatively small sets of training data, making it possible to generate
training data using highly accurate methods at a reasonable computational cost.
We present our approach, the forms of the discovered models, and assessments of
their transferability, accuracy and speed
Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals
An integral scheme for the efficient evaluation of two-center integrals over
contracted solid harmonic Gaussian functions is presented. Integral expressions
are derived for local operators that depend on the position vector of one of
the two Gaussian centers. These expressions are then used to derive the formula
for three-index overlap integrals where two of the three Gaussians are located
at the same center. The efficient evaluation of the latter is essential for
local resolution-of-the-identity techniques that employ an overlap metric. We
compare the performance of our integral scheme to the widely used Cartesian
Gaussian-based method of Obara and Saika (OS). Non-local interaction potentials
such as standard Coulomb, modified Coulomb and Gaussian-type operators, that
occur in range-separated hybrid functionals, are also included in the
performance tests. The speed-up with respect to the OS scheme is up to three
orders of magnitude for both, integrals and their derivatives. In particular,
our method is increasingly efficient for large angular momenta and highly
contracted basis sets.Comment: 18 pages, 2 figures; accepted manuscript. v2: supplementary material
include
A parallel algorithm for Hamiltonian matrix construction in electron-molecule collision calculations: MPI-SCATCI
Construction and diagonalization of the Hamiltonian matrix is the
rate-limiting step in most low-energy electron -- molecule collision
calculations. Tennyson (J Phys B, 29 (1996) 1817) implemented a novel algorithm
for Hamiltonian construction which took advantage of the structure of the
wavefunction in such calculations. This algorithm is re-engineered to make use
of modern computer architectures and the use of appropriate diagonalizers is
considered. Test calculations demonstrate that significant speed-ups can be
gained using multiple CPUs. This opens the way to calculations which consider
higher collision energies, larger molecules and / or more target states. The
methodology, which is implemented as part of the UK molecular R-matrix codes
(UKRMol and UKRMol+) can also be used for studies of bound molecular Rydberg
states, photoionisation and positron-molecule collisions.Comment: Write up of a computer program MPI-SCATCI Computer Physics
Communications, in pres
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
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