58,822 research outputs found
Data-adaptive harmonic spectra and multilayer Stuart-Landau models
Harmonic decompositions of multivariate time series are considered for which
we adopt an integral operator approach with periodic semigroup kernels.
Spectral decomposition theorems are derived that cover the important cases of
two-time statistics drawn from a mixing invariant measure.
The corresponding eigenvalues can be grouped per Fourier frequency, and are
actually given, at each frequency, as the singular values of a cross-spectral
matrix depending on the data. These eigenvalues obey furthermore a variational
principle that allows us to define naturally a multidimensional power spectrum.
The eigenmodes, as far as they are concerned, exhibit a data-adaptive character
manifested in their phase which allows us in turn to define a multidimensional
phase spectrum.
The resulting data-adaptive harmonic (DAH) modes allow for reducing the
data-driven modeling effort to elemental models stacked per frequency, only
coupled at different frequencies by the same noise realization. In particular,
the DAH decomposition extracts time-dependent coefficients stacked by Fourier
frequency which can be efficiently modeled---provided the decay of temporal
correlations is sufficiently well-resolved---within a class of multilayer
stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators.
Applications to the Lorenz 96 model and to a stochastic heat equation driven
by a space-time white noise, are considered. In both cases, the DAH
decomposition allows for an extraction of spatio-temporal modes revealing key
features of the dynamics in the embedded phase space. The multilayer
Stuart-Landau models (MSLMs) are shown to successfully model the typical
patterns of the corresponding time-evolving fields, as well as their statistics
of occurrence.Comment: 26 pages, double columns; 15 figure
Multidimensional approximation of nonlinear dynamical systems
A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems
Molecular Model of Dynamic Social Network Based on E-mail communication
In this work we consider an application of physically inspired sociodynamical model to the modelling of the evolution of email-based social network. Contrary to the standard approach of sociodynamics, which assumes expressing of system dynamics with heuristically defined simple rules, we postulate the inference of these rules from the real data and their application within a dynamic molecular model. We present how to embed the n-dimensional social space in Euclidean one. Then, inspired by the Lennard-Jones potential, we define a data-driven social potential function and apply the resultant force to a real e-mail communication network in a course of a molecular simulation, with network nodes taking on the role of interacting particles. We discuss all steps of the modelling process, from data preparation, through embedding and the molecular simulation itself, to transformation from the embedding space back to a graph structure. The conclusions, drawn from examining the resultant networks in stable, minimum-energy states, emphasize the role of the embedding process projecting the non–metric social graph into the Euclidean space, the significance of the unavoidable loss of information connected with this procedure and the resultant preservation of global rather than local properties of the initial network. We also argue applicability of our method to some classes of problems, while also signalling the areas which require further research in order to expand this applicability domain
Multidimensional relativistic MHD simulations of Pulsar Wind Nebulae: dynamics and emission
Pulsar Wind Nebulae, and the Crab nebula in particular, are the best cosmic
laboratories to investigate the dynamics of magnetized relativistic outflows
and particle acceleration up to PeV energies. Multidimensional MHD modeling by
means of numerical simulations has been very successful at reproducing, to the
very finest details, the innermost structure of these synchrotron emitting
nebulae, as observed in the X-rays. Therefore, the comparison between the
simulated source and observations can be used as a powerful diagnostic tool to
probe the physical conditions in pulsar winds, like their composition,
magnetization, and degree of anisotropy. However, in spite of the wealth of
observations and of the accuracy of current MHD models, the precise mechanisms
for magnetic field dissipation and for the acceleration of the non-thermal
emitting particles are mysteries still puzzling theorists to date. Here we
review the methodologies of the computational approach to the modeling of
Pulsar Wind Nebulae, discussing the most relevant results and the recent
progresses achieved in this fascinating field of high-energy astrophysics.Comment: 29 pages review, preliminary version. To appear in the book
"Modelling Nebulae" edited by D. Torres for Springer, based on the invited
contributions to the workshop held in Sant Cugat (Barcelona), June 14-17,
201
WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations
WavePacket is an open-source program package for the numerical simulation of
quantum-mechanical dynamics. It can be used to solve time-independent or
time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one
or more dimensions. Also coupled equations can be treated, which allows to
simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation.
Optionally accounting for the interaction with external electric fields within
the semiclassical dipole approximation, WavePacket can be used to simulate
experiments involving tailored light pulses in photo-induced physics or
chemistry.The graphical capabilities allow visualization of quantum dynamics
'on the fly', including Wigner phase space representations. Being easy to use
and highly versatile, WavePacket is well suited for the teaching of quantum
mechanics as well as for research projects in atomic, molecular and optical
physics or in physical or theoretical chemistry.The present Part I deals with
the description of closed quantum systems in terms of Schr\"odinger equations.
The emphasis is on discrete variable representations for spatial discretization
as well as various techniques for temporal discretization.The upcoming Part II
will focus on open quantum systems and dimension reduction; it also describes
the codes for optimal control of quantum dynamics.The present work introduces
the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge
platform, where extensive Wiki-documentation as well as worked-out
demonstration examples can be found
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