129 research outputs found
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
Efficient GPU implementation of a Boltzmann‑Schrödinger‑Poisson solver for the simulation of nanoscale DG MOSFETs
81–102, 2019) describes an efficient and accurate solver for nanoscale DG MOSFETs
through a deterministic Boltzmann-Schrödinger-Poisson model with seven
electron–phonon scattering mechanisms on a hybrid parallel CPU/GPU platform.
The transport computational phase, i.e. the time integration of the Boltzmann equations,
was ported to the GPU using CUDA extensions, but the computation of the
system’s eigenstates, i.e. the solution of the Schrödinger-Poisson block, was parallelized
only using OpenMP due to its complexity. This work fills the gap by describing
a port to GPU for the solver of the Schrödinger-Poisson block. This new proposal
implements on GPU a Scheduled Relaxation Jacobi method to solve the sparse linear
systems which arise in the 2D Poisson equation. The 1D Schrödinger equation
is solved on GPU by adapting a multi-section iteration and the Newton-Raphson
algorithm to approximate the energy levels, and the Inverse Power Iterative Method
is used to approximate the wave vectors. We want to stress that this solver for the
Schrödinger-Poisson block can be thought as a module independent of the transport
phase (Boltzmann) and can be used for solvers using different levels of description
for the electrons; therefore, it is of particular interest because it can be adapted to
other macroscopic, hence faster, solvers for confined devices exploited at industrial
level.Project PID2020-117846GB-I00 funded by the Spanish Ministerio de Ciencia
e InnovaciónProject A-TIC-344-UGR20 funded by European
Regional Development Fund
An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues --- a Study for the Dirac Operator in SU(2) Lattice QCD
The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with
controlled numerical errors by a conjugate gradient (CG) method. This CG
algorithm is accelerated by alternating it with exact diagonalisations in the
subspace spanned by the numerically computed eigenvectors. We study this
combined algorithm in case of the Dirac operator with (dynamical) Wilson
fermions in four-dimensional \SUtwo gauge fields. The algorithm is
numerically very stable and can be parallelized in an efficient way. On
lattices of sizes an acceleration of the pure CG method by a factor
of~ is found.Comment: 25 pages, uuencoded tar-compressed .ps-fil
Approximating spectral densities of large matrices
In physics, it is sometimes desirable to compute the so-called \emph{Density
Of States} (DOS), also known as the \emph{spectral density}, of a real
symmetric matrix . The spectral density can be viewed as a probability
density distribution that measures the likelihood of finding eigenvalues near
some point on the real line. The most straightforward way to obtain this
density is to compute all eigenvalues of . But this approach is generally
costly and wasteful, especially for matrices of large dimension. There exists
alternative methods that allow us to estimate the spectral density function at
much lower cost. The major computational cost of these methods is in
multiplying with a number of vectors, which makes them appealing for
large-scale problems where products of the matrix with arbitrary vectors
are relatively inexpensive. This paper defines the problem of estimating the
spectral density carefully, and discusses how to measure the accuracy of an
approximate spectral density. It then surveys a few known methods for
estimating the spectral density, and proposes some new variations of existing
methods. All methods are discussed from a numerical linear algebra point of
view
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