2,453 research outputs found
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
BBGKY Dynamics: from Localization to Pattern Formation
A fast and efficient numerical-analytical approach is proposed for modeling
complex behaviour in the BBGKY--hierarchy of kinetic equations. Our
calculations are based on variational and multiresolution approaches in the
basis of polynomial tensor algebras of generalized coherent states/wavelets. We
construct the representation for hierarchy of reduced distribution functions
via the multiscale decomposition in highly-localized eigenmodes. Numerical
modeling shows the creation of various internal structures from localized
modes, which are related to localized or chaotic type of behaviour and the
corresponding patterns (waveletons) formation. The localized pattern is a model
for energy confinement state (fusion) in plasma.Comment: 14 pages, 3 figures, ws-procs9x6.cls, presented at Workshop "Progress
in Nonequilibrium Greens Functions", Dresden, Germany, August 19-23, 200
Innovative Second-Generation Wavelets Construction With Recurrent Neural Networks for Solar Radiation Forecasting
Solar radiation prediction is an important challenge for the electrical
engineer because it is used to estimate the power developed by commercial
photovoltaic modules. This paper deals with the problem of solar radiation
prediction based on observed meteorological data. A 2-day forecast is obtained
by using novel wavelet recurrent neural networks (WRNNs). In fact, these WRNNS
are used to exploit the correlation between solar radiation and
timescale-related variations of wind speed, humidity, and temperature. The
input to the selected WRNN is provided by timescale-related bands of wavelet
coefficients obtained from meteorological time series. The experimental setup
available at the University of Catania, Italy, provided this information. The
novelty of this approach is that the proposed WRNN performs the prediction in
the wavelet domain and, in addition, also performs the inverse wavelet
transform, giving the predicted signal as output. The obtained simulation
results show a very low root-mean-square error compared to the results of the
solar radiation prediction approaches obtained by hybrid neural networks
reported in the recent literature
Wavelets and Fast Numerical Algorithms
Wavelet based algorithms in numerical analysis are similar to other transform
methods in that vectors and operators are expanded into a basis and the
computations take place in this new system of coordinates. However, due to the
recursive definition of wavelets, their controllable localization in both space
and wave number (time and frequency) domains, and the vanishing moments
property, wavelet based algorithms exhibit new and important properties.
For example, the multiresolution structure of the wavelet expansions brings
about an efficient organization of transformations on a given scale and of
interactions between different neighbouring scales. Moreover, wide classes of
operators which naively would require a full (dense) matrix for their numerical
description, have sparse representations in wavelet bases. For these operators
sparse representations lead to fast numerical algorithms, and thus address a
critical numerical issue.
We note that wavelet based algorithms provide a systematic generalization of
the Fast Multipole Method (FMM) and its descendents.
These topics will be the subject of the lecture. Starting from the notion of
multiresolution analysis, we will consider the so-called non-standard form
(which achieves decoupling among the scales) and the associated fast numerical
algorithms. Examples of non-standard forms of several basic operators (e.g.
derivatives) will be computed explicitly.Comment: 32 pages, uuencoded tar-compressed LaTeX file. Uses epsf.sty (see
`macros'
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