24,720 research outputs found
Basic Types of Coarse-Graining
We consider two basic types of coarse-graining: the Ehrenfests'
coarse-graining and its extension to a general principle of non-equilibrium
thermodynamics, and the coarse-graining based on uncertainty of dynamical
models and Epsilon-motions (orbits). Non-technical discussion of basic notions
and main coarse-graining theorems are presented: the theorem about entropy
overproduction for the Ehrenfests' coarse-graining and its generalizations,
both for conservative and for dissipative systems, and the theorems about
stable properties and the Smale order for Epsilon-motions of general dynamical
systems including structurally unstable systems. Computational kinetic models
of macroscopic dynamics are considered. We construct a theoretical basis for
these kinetic models using generalizations of the Ehrenfests' coarse-graining.
General theory of reversible regularization and filtering semigroups in
kinetics is presented, both for linear and non-linear filters. We obtain
explicit expressions and entropic stability conditions for filtered equations.
A brief discussion of coarse-graining by rounding and by small noise is also
presented.Comment: 60 pgs, 11 figs., includes new analysis of coarse-graining by
filtering. A talk given at the research workshop: "Model Reduction and
Coarse-Graining Approaches for Multiscale Phenomena," University of
Leicester, UK, August 24-26, 200
Effects of coarse-graining on the scaling behavior of long-range correlated and anti-correlated signals
We investigate how various coarse-graining methods affect the scaling
properties of long-range power-law correlated and anti-correlated signals,
quantified by the detrended fluctuation analysis. Specifically, for
coarse-graining in the magnitude of a signal, we consider (i) the Floor, (ii)
the Symmetry and (iii) the Centro-Symmetry coarse-graining methods. We find,
that for anti-correlated signals coarse-graining in the magnitude leads to a
crossover to random behavior at large scales, and that with increasing the
width of the coarse-graining partition interval this crossover moves
to intermediate and small scales. In contrast, the scaling of positively
correlated signals is less affected by the coarse-graining, with no observable
changes when a crossover appears at small
scales and moves to intermediate and large scales with increasing . For
very rough coarse-graining () based on the Floor and Symmetry
methods, the position of the crossover stabilizes, in contrast to the
Centro-Symmetry method where the crossover continuously moves across scales and
leads to a random behavior at all scales, thus indicating a much stronger
effect of the Centro-Symmetry compared to the Floor and the Symmetry methods.
For coarse-graining in time, where data points are averaged in non-overlapping
time windows, we find that the scaling for both anti-correlated and positively
correlated signals is practically preserved. The results of our simulations are
useful for the correct interpretation of the correlation and scaling properties
of symbolic sequences.Comment: 19 pages, 13 figure
Diffusion-Based Coarse Graining in Hybrid Continuum-Discrete Solvers: Theoretical Formulation and A Priori Tests
Coarse graining is an important ingredient in many multi-scale
continuum-discrete solvers such as CFD--DEM (computational fluid
dynamics--discrete element method) solvers for dense particle-laden flows.
Although CFD--DEM solvers have become a mature technique that is widely used in
multiphase flow research and industrial flow simulations, a flexible and
easy-to-implement coarse graining algorithm that can work with CFD solvers of
arbitrary meshes is still lacking. In this work, we proposed a new coarse
graining algorithm for continuum--discrete solvers for dense particle-laden
flows based on solving a transient diffusion equation. Via theoretical analysis
we demonstrated that the proposed method is equivalent to the statistical
kernel method with a Gaussian kernel, but the current method is much more
straightforward to implement in CFD--DEM solvers. \textit{A priori} numerical
tests were performed to obtain the solid volume fraction fields based on given
particle distributions, the results obtained by using the proposed algorithm
were compared with those from other coarse graining methods in the literature
(e.g., the particle centroid method, the divided particle volume method, and
the two-grid formulation). The numerical tests demonstrated that the proposed
coarse graining procedure based on solving diffusion equations is theoretically
sound, easy to implement and parallelize in general CFD solvers, and has
improved mesh-convergence characteristics compared with existing coarse
graining methods. The diffusion-based coarse graining method has been
implemented into a CFD--DEM solver, the results of which are presented in a
separate work (R. Sun and H. Xiao, Diffusion-based coarse graining in hybrid
continuum-discrete solvers: Application in CFD-DEM solvers for particle laden
flows)
Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics
In molecular dynamics and sampling of high dimensional Gibbs measures
coarse-graining is an important technique to reduce the dimensionality of the
problem. We will study and quantify the coarse-graining error between the
coarse-grained dynamics and an effective dynamics. The effective dynamics is a
Markov process on the coarse-grained state space obtained by a closure
procedure from the coarse-grained coefficients. We obtain error estimates both
in relative entropy and Wasserstein distance, for both Langevin and overdamped
Langevin dynamics. The approach allows for vectorial coarse-graining maps.
Hereby, the quality of the chosen coarse-graining is measured by certain
functional inequalities encoding the scale separation of the Gibbs measure. The
method is based on error estimates between solutions of (kinetic) Fokker-Planck
equations in terms of large-deviation rate functionals
Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems
The primary objective of this work is to develop coarse-graining schemes for
stochastic many-body microscopic models and quantify their effectiveness in
terms of a priori and a posteriori error analysis. In this paper we focus on
stochastic lattice systems of interacting particles at equilibrium. %such as
Ising-type models. The proposed algorithms are derived from an initial
coarse-grained approximation that is directly computable by Monte Carlo
simulations, and the corresponding numerical error is calculated using the
specific relative entropy between the exact and approximate coarse-grained
equilibrium measures. Subsequently we carry out a cluster expansion around this
first-and often inadequate-approximation and obtain more accurate
coarse-graining schemes. The cluster expansions yield also sharp a posteriori
error estimates for the coarse-grained approximations that can be used for the
construction of adaptive coarse-graining methods. We present a number of
numerical examples that demonstrate that the coarse-graining schemes developed
here allow for accurate predictions of critical behavior and hysteresis in
systems with intermediate and long-range interactions. We also present examples
where they substantially improve predictions of earlier coarse-graining schemes
for short-range interactions.Comment: 37 pages, 8 figure
Spectral coarse graining for random walk in bipartite networks
Many real-world networks display a natural bipartite structure, while
analyzing or visualizing large bipartite networks is one of the most
challenges. As a result, it is necessary to reduce the complexity of large
bipartite systems and preserve the functionality at the same time. We observe,
however, the existing coarse graining methods for binary networks fail to work
in the bipartite networks. In this paper, we use the spectral analysis to
design a coarse graining scheme specifically for bipartite networks and keep
their random walk properties unchanged. Numerical analysis on artificial and
real-world bipartite networks indicates that our coarse graining scheme could
obtain much smaller networks from large ones, keeping most of the relevant
spectral properties. Finally, we further validate the coarse graining method by
directly comparing the mean first passage time between the original network and
the reduced one.Comment: 7 pages, 3 figure
Spectral coarse-graining of complex networks
Reducing the complexity of large systems described as complex networks is key
to understand them and a crucial issue is to know which properties of the
initial system are preserved in the reduced one. Here we use random walks to
design a coarse-graining scheme for complex networks. By construction the
coarse-graining preserves the slow modes of the walk, while reducing
significantly the size and the complexity of the network. In this sense our
coarse-graining allows to approximate large networks by smaller ones, keeping
most of their relevant spectral properties.Comment: 4 pages, 2 figure
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