9,099 research outputs found
Gradient type optimization methods for electronic structure calculations
The density functional theory (DFT) in electronic structure calculations can
be formulated as either a nonlinear eigenvalue or direct minimization problem.
The most widely used approach for solving the former is the so-called
self-consistent field (SCF) iteration. A common observation is that the
convergence of SCF is not clear theoretically while approaches with convergence
guarantee for solving the latter are often not competitive to SCF numerically.
In this paper, we study gradient type methods for solving the direct
minimization problem by constructing new iterations along the gradient on the
Stiefel manifold. Global convergence (i.e., convergence to a stationary point
from any initial solution) as well as local convergence rate follows from the
standard theory for optimization on manifold directly. A major computational
advantage is that the computation of linear eigenvalue problems is no longer
needed. The main costs of our approaches arise from the assembling of the total
energy functional and its gradient and the projection onto the manifold. These
tasks are cheaper than eigenvalue computation and they are often more suitable
for parallelization as long as the evaluation of the total energy functional
and its gradient is efficient. Numerical results show that they can outperform
SCF consistently on many practically large systems.Comment: 24 pages, 11 figures, 59 references, and 1 acknowledgement
Adaptive Finite Element Approximations for Kohn-Sham Models
The Kohn-Sham equation is a powerful, widely used approach for computation of
ground state electronic energies and densities in chemistry, materials science,
biology, and nanosciences. In this paper, we study the adaptive finite element
approximations for the Kohn-Sham model. Based on the residual type a posteriori
error estimators proposed in this paper, we introduce an adaptive finite
element algorithm with a quite general marking strategy and prove the
convergence of the adaptive finite element approximations. Using D{\" o}rfler's
marking strategy, we then get the convergence rate and quasi-optimal
complexity. We also carry out several typical numerical experiments that not
only support our theory,but also show the robustness and efficiency of the
adaptive finite element computations in electronic structure calculations.Comment: 38pages, 7figure
A Direct Multigrid Poisson Solver for Oct-Tree Adaptive Meshes
We describe a finite-volume method for solving the Poisson equation on
oct-tree adaptive meshes using direct solvers for individual mesh blocks. The
method is a modified version of the method presented by Huang and Greengard
(2000), which works with finite-difference meshes and does not allow for shared
boundaries between refined patches. Our algorithm is implemented within the
FLASH code framework and makes use of the PARAMESH library, permitting
efficient use of parallel computers. We describe the algorithm and present test
results that demonstrate its accuracy.Comment: 10 pages, 6 figures, accepted by the Astrophysical Journal; minor
revisions in response to referee's comments; added char
MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation
MADNESS (multiresolution adaptive numerical environment for scientific
simulation) is a high-level software environment for solving integral and
differential equations in many dimensions that uses adaptive and fast harmonic
analysis methods with guaranteed precision based on multiresolution analysis
and separated representations. Underpinning the numerical capabilities is a
powerful petascale parallel programming environment that aims to increase both
programmer productivity and code scalability. This paper describes the features
and capabilities of MADNESS and briefly discusses some current applications in
chemistry and several areas of physics
RAM: A Relativistic Adaptive Mesh Refinement Hydrodynamics Code
We have developed a new computer code, RAM, to solve the conservative
equations of special relativistic hydrodynamics (SRHD) using adaptive mesh
refinement (AMR) on parallel computers. We have implemented a
characteristic-wise, finite difference, weighted essentially non-oscillatory
(WENO) scheme using the full characteristic decomposition of the SRHD equations
to achieve fifth-order accuracy in space. For time integration we use the
method of lines with a third-order total variation diminishing (TVD)
Runge-Kutta scheme. We have also implemented fourth and fifth order Runge-Kutta
time integration schemes for comparison. The implementation of AMR and
parallelization is based on the FLASH code. RAM is modular and includes the
capability to easily swap hydrodynamics solvers, reconstruction methods and
physics modules. In addition to WENO we have implemented a finite volume module
with the piecewise parabolic method (PPM) for reconstruction and the modified
Marquina approximate Riemann solver to work with TVD Runge-Kutta time
integration. We examine the difficulty of accurately simulating shear flows in
numerical relativistic hydrodynamics codes. We show that under-resolved
simulations of simple test problems with transverse velocity components produce
incorrect results and demonstrate the ability of RAM to correctly solve these
problems. RAM has been tested in one, two and three dimensions and in
Cartesian, cylindrical and spherical coordinates. We have demonstrated
fifth-order accuracy for WENO in one and two dimensions and performed detailed
comparison with other schemes for which we show significantly lower convergence
rates. Extensive testing is presented demonstrating the ability of RAM to
address challenging open questions in relativistic astrophysics.Comment: ApJS in press, 21 pages including 18 figures (6 color figures
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