45 research outputs found
Efficient simulations of Hartree--Fock equations by an accelerated gradient descent method
We develop convergence acceleration procedures that enable a gradient
descent-type iteration method to efficiently simulate Hartree--Fock equations
for atoms interacting both with each other and with an external potential. Our
development focuses on three aspects: (i) optimization of a parameter in the
preconditioning operator; (ii) adoption of a technique that eliminates the
slowest-decaying mode to the case of many equations (describing many atoms);
and (iii) a novel extension of the above technique that allows one to eliminate
multiple modes simultaneously. We illustrate performance of the numerical
method for the 2D model of the first layer of helium atoms above a graphene
sheet. We demonstrate that incorporation of aspects (i) and (ii) above into the
``plain" gradient descent method accelerates it by at least two orders of
magnitude, and often by much more. Aspect (iii) -- a multiple-mode elimination
-- may bring further improvement to the convergence rate compared to aspect
(ii), the single-mode elimination. Both single- and multiple-mode elimination
techniques are shown to significantly outperform the well-known Anderson
Acceleration. We believe that our acceleration techniques can also be gainfully
employed by other numerical methods, especially those handling hard-core-type
interaction potentials.Comment: main text (39 pages); supplement appended (7 pages
Modeling the desired direction in a force-based model for pedestrian dynamics
We introduce an enhanced model based on the generalized centrifugal force
model. Furthermore, the desired direction of pedestrians is investigated. A new
approach leaning on the well-known concept of static and dynamic floor-fields
in cellular automata is presented. Numerical results of the model are presented
and compared with empirical data.Comment: 14 pages 11 figures, submitted to TGF'1
Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schr\"odinger equation
We consider in detail the self-trapping of a soliton from a wave pulse that
passes from a defocussing region into a focussing one in a spatially
inhomogeneous nonlinear waveguide, described by a nonlinear Schrodinger
equation in which the dispersion coefficient changes its sign from normal to
anomalous. The model has direct applications to dispersion-decreasing nonlinear
optical fibers, and to natural waveguides for internal waves in the ocean. It
is found that, depending on the (conserved) energy and (nonconserved) mass of
the initial pulse, four qualitatively different outcomes of the pulse
transformation are possible: decay into radiation; self-trapping into a single
soliton; formation of a breather; and formation of a pair of counterpropagating
solitons. A corresponding chart is drawn on a parametric plane, which
demonstrates some unexpected features. In particular, it is found that any kind
of soliton(s) (including the breather and counterpropagating pair) eventually
decays into pure radiation with the increase of the energy, the initial mass
being kept constant. It is also noteworthy that a virtually direct transition
from a single soliton into a pair of symmetric counterpropagating ones seems
possible. An explanation for these features is proposed. In two cases when
analytical approximations apply, viz., a simple perturbation theory for broad
initial pulses, or the variational approximation for narrow ones, comparison
with the direct simulations shows reasonable agreement.Comment: 18 pages, 10 figures, 1 table. Phys. Rev. E, in pres
On the boundary of the dispersion-managed soliton existence
A breathing soliton-like structure in dispersion-managed (DM) optical fiber
system is studied. It is proven that for negative average dispersion the
breathing soliton is forbidden provided that a modulus of average dispersion
exceed a threshold which depends on the soliton amplitude.Comment: LaTeX, 8 pages, to appear in JETP Lett. 72, #3 (2000
On non-local variational problems with lack of compactness related to non-linear optics
We give a simple proof of existence of solutions of the dispersion manage-
ment and diffraction management equations for zero average dispersion,
respectively diffraction. These solutions are found as maximizers of non-linear
and non-local vari- ational problems which are invariant under a large
non-compact group. Our proof of existence of maximizer is rather direct and
avoids the use of Lions' concentration compactness argument or Ekeland's
variational principle.Comment: 30 page
Hamiltonian form and solitary waves of the spatial Dysthe equations
The spatial Dysthe equations describe the envelope evolution of the
free-surface and potential of gravity waves in deep waters. Their Hamiltonian
structure and new invariants are unveiled by means of a gauge transformation to
a new canonical form of the evolution equations. An accurate Fourier-type
spectral scheme is used to solve for the wave dynamics and validate the new
conservation laws, which are satisfied up to machine precision. Traveling waves
are numerically constructed using the Petviashvili method. It is shown that
their collision appears inelastic, suggesting the non-integrability of the
Dysthe equations.Comment: 6 pages, 9 figures. Other author's papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutykh