1,537 research outputs found
Spatiospectral concentration on a sphere
We pose and solve the analogue of Slepian's time-frequency concentration
problem on the surface of the unit sphere to determine an orthogonal family of
strictly bandlimited functions that are optimally concentrated within a closed
region of the sphere, or, alternatively, of strictly spacelimited functions
that are optimally concentrated within the spherical harmonic domain. Such a
basis of simultaneously spatially and spectrally concentrated functions should
be a useful data analysis and representation tool in a variety of geophysical
and planetary applications, as well as in medical imaging, computer science,
cosmology and numerical analysis. The spherical Slepian functions can be found
either by solving an algebraic eigenvalue problem in the spectral domain or by
solving a Fredholm integral equation in the spatial domain. The associated
eigenvalues are a measure of the spatiospectral concentration. When the
concentration region is an axisymmetric polar cap the spatiospectral projection
operator commutes with a Sturm-Liouville operator; this enables the
eigenfunctions to be computed extremely accurately and efficiently, even when
their area-bandwidth product, or Shannon number, is large. In the asymptotic
limit of a small concentration region and a large spherical harmonic bandwidth
the spherical concentration problem approaches its planar equivalent, which
exhibits self-similarity when the Shannon number is kept invariant.Comment: 48 pages, 17 figures. Submitted to SIAM Review, August 24th, 200
An application of interpolating scaling functions to wave packet propagation
Wave packet propagation in the basis of interpolating scaling functions (ISF)
is studied. The ISF are well known in the multiresolution analysis based on
spline biorthogonal wavelets. The ISF form a cardinal basis set corresponding
to an equidistantly spaced grid. They have compact support of the size
determined by the underlying interpolating polynomial that is used to generate
ISF. In this basis the potential energy matrix is diagonal and the kinetic
energy matrix is sparse and, in the 1D case, has a band-diagonal structure. An
important feature of the basis is that matrix elements of a Hamiltonian are
exactly computed by means of simple algebraic transformations efficiently
implemented numerically. Therefore the number of grid points and the order of
the underlying interpolating polynomial can easily be varied allowing one to
approach the accuracy of pseudospectral methods in a regular manner, similar to
high order finite difference methods. The results of numerical simulations of
an H+H_2 collinear collision show that the ISF provide one with an accurate and
efficient representation for use in the wave packet propagation method.Comment: plain Latex, 11 pages, 4 figures attached in the JPEG forma
Two-level Chebyshev filter based complementary subspace method: pushing the envelope of large-scale electronic structure calculations
We describe a novel iterative strategy for Kohn-Sham density functional
theory calculations aimed at large systems (> 1000 electrons), applicable to
metals and insulators alike. In lieu of explicit diagonalization of the
Kohn-Sham Hamiltonian on every self-consistent field (SCF) iteration, we employ
a two-level Chebyshev polynomial filter based complementary subspace strategy
to: 1) compute a set of vectors that span the occupied subspace of the
Hamiltonian; 2) reduce subspace diagonalization to just partially occupied
states; and 3) obtain those states in an efficient, scalable manner via an
inner Chebyshev-filter iteration. By reducing the necessary computation to just
partially occupied states, and obtaining these through an inner Chebyshev
iteration, our approach reduces the cost of large metallic calculations
significantly, while eliminating subspace diagonalization for insulating
systems altogether. We describe the implementation of the method within the
framework of the Discontinuous Galerkin (DG) electronic structure method and
show that this results in a computational scheme that can effectively tackle
bulk and nano systems containing tens of thousands of electrons, with chemical
accuracy, within a few minutes or less of wall clock time per SCF iteration on
large-scale computing platforms. We anticipate that our method will be
instrumental in pushing the envelope of large-scale ab initio molecular
dynamics. As a demonstration of this, we simulate a bulk silicon system
containing 8,000 atoms at finite temperature, and obtain an average SCF step
wall time of 51 seconds on 34,560 processors; thus allowing us to carry out 1.0
ps of ab initio molecular dynamics in approximately 28 hours (of wall time).Comment: Resubmitted version (version 2
Chebyshev polynomial filtered subspace iteration in the Discontinuous Galerkin method for large-scale electronic structure calculations
The Discontinuous Galerkin (DG) electronic structure method employs an
adaptive local basis (ALB) set to solve the Kohn-Sham equations of density
functional theory (DFT) in a discontinuous Galerkin framework. The adaptive
local basis is generated on-the-fly to capture the local material physics, and
can systematically attain chemical accuracy with only a few tens of degrees of
freedom per atom. A central issue for large-scale calculations, however, is the
computation of the electron density (and subsequently, ground state properties)
from the discretized Hamiltonian in an efficient and scalable manner. We show
in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can
be used to address this issue and push the envelope in large-scale materials
simulations in a discontinuous Galerkin framework. We describe how the subspace
filtering steps can be performed in an efficient and scalable manner using a
two-dimensional parallelization scheme, thanks to the orthogonality of the DG
basis set and block-sparse structure of the DG Hamiltonian matrix. The
on-the-fly nature of the ALBs requires additional care in carrying out the
subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI
approach in calculations of large-scale two-dimensional graphene sheets and
bulk three-dimensional lithium-ion electrolyte systems. Employing 55,296
computational cores, the time per self-consistent field iteration for a sample
of the bulk 3D electrolyte containing 8,586 atoms is 90 seconds, and the time
for a graphene sheet containing 11,520 atoms is 75 seconds.Comment: Submitted to The Journal of Chemical Physic
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