56 research outputs found
Efficient iterative method for solving the Dirac-Kohn-Sham density functional theory
We present for the first time an efficient iterative method to directly solve
the four-component Dirac-Kohn-Sham (DKS) density functional theory. Due to the
existence of the negative energy continuum in the DKS operator, the existing
iterative techniques for solving the Kohn-Sham systems cannot be efficiently
applied to solve the DKS systems. The key component of our method is a novel
filtering step (F) which acts as a preconditioner in the framework of the
locally optimal block preconditioned conjugate gradient (LOBPCG) method. The
resulting method, dubbed the LOBPCG-F method, is able to compute the desired
eigenvalues and eigenvectors in the positive energy band without computing any
state in the negative energy band. The LOBPCG-F method introduces mild extra
cost compared to the standard LOBPCG method and can be easily implemented. We
demonstrate our method in the pseudopotential framework with a planewave basis
set which naturally satisfies the kinetic balance prescription. Numerical
results for Pt, Au, TlF, and BiSe indicate that the
LOBPCG-F method is a robust and efficient method for investigating the
relativistic effect in systems containing heavy elements.Comment: 31 pages, 5 figure
Numerical methods for nonlinear Dirac equation
This paper presents a review of the current state-of-the-art of numerical
methods for nonlinear Dirac (NLD) equation. Several methods are extendedly
proposed for the (1+1)-dimensional NLD equation with the scalar and vector
self-interaction and analyzed in the way of the accuracy and the time
reversibility as well as the conservation of the discrete charge, energy and
linear momentum. Those methods are the Crank-Nicolson (CN) schemes, the
linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme, a
semi-implicit finite difference scheme, and the exponential operator splitting
(OS) schemes. The nonlinear subproblems resulted from the OS schemes are
analytically solved by fully exploiting the local conservation laws of the NLD
equation. The effectiveness of the various numerical methods, with special
focus on the error growth and the computational cost, is illustrated on two
numerical experiments, compared to two high-order accurate Runge-Kutta
discontinuous Galerkin methods. Theoretical and numerical comparisons show that
the high-order accurate OS schemes may compete well with other numerical
schemes discussed here in terms of the accuracy and the efficiency. A
fourth-order accurate OS scheme is further applied to investigating the
interaction dynamics of the NLD solitary waves under the scalar and vector
self-interaction. The results show that the interaction dynamics of two NLD
solitary waves depend on the exponent power of the self-interaction in the NLD
equation; collapse happens after collision of two equal one-humped NLD solitary
waves under the cubic vector self-interaction in contrast to no collapse
scattering for corresponding quadric case.Comment: 39 pages, 13 figure
A simple iterative algorithm for maxcut
We propose a simple iterative (SI) algorithm for the maxcut problem through
fully using an equivalent continuous formulation. It does not need rounding at
all and has advantages that all subproblems have explicit analytic solutions,
the cut values are monotonically updated and the iteration points converge to a
local optima in finite steps via an appropriate subgradient selection.
Numerical experiments on G-set demonstrate the performance. In particular, the
ratios between the best cut values achieved by SI and the best known ones are
at least and can be further improved to at least by a
preliminary attempt to break out of local optima.Comment: 30 pages, 1 figure. Subgradient selection, cost analysis and local
breakout are adde
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