6,762 research outputs found
Multiscale theory of turbulence in wavelet representation
We present a multiscale description of hydrodynamic turbulence in
incompressible fluid based on a continuous wavelet transform (CWT) and a
stochastic hydrodynamics formalism. Defining the stirring random force by the
correlation function of its wavelet components, we achieve the cancellation of
loop divergences in the stochastic perturbation expansion. An extra
contribution to the energy transfer from large to smaller scales is considered.
It is shown that the Kolmogorov hypotheses are naturally reformulated in
multiscale formalism. The multiscale perturbation theory and statistical
closures based on the wavelet decomposition are constructed.Comment: LaTeX, 27 pages, 3 eps figure
Scale-Dependent Functions, Stochastic Quantization and Renormalization
We consider a possibility to unify the methods of regularization, such as the
renormalization group method, stochastic quantization etc., by the extension of
the standard field theory of the square-integrable functions to the theory of functions that depend on coordinate
and resolution . In the simplest case such field theory turns out to be a
theory of fields defined on the affine group ,
, which consists of dilations and translation of
Euclidean space. The fields are constructed using the
continuous wavelet transform. The parameters of the theory can explicitly
depend on the resolution . The proper choice of the scale dependence
makes such theory free of divergences by construction.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Generalized chronotaxic systems: time-dependent oscillatory dynamics stable under continuous perturbation
Chronotaxic systems represent deterministic nonautonomous oscillatory systems
which are capable of resisting continuous external perturbations while having a
complex time-dependent dynamics. Until their recent introduction in \emph{Phys.
Rev. Lett.} \textbf{111}, 024101 (2013) chronotaxic systems had often been
treated as stochastic, inappropriately, and the deterministic component had
been ignored. While the previous work addressed the case of the decoupled
amplitude and phase dynamics, in this paper we develop a generalized theory of
chronotaxic systems where such decoupling is not required. The theory presented
is based on the concept of a time-dependent point attractor or a driven steady
state and on the contraction theory of dynamical systems. This simplifies the
analysis of chronotaxic systems and makes possible the identification of
chronotaxic systems with time-varying parameters. All types of chronotaxic
dynamics are classified and their properties are discussed using the
nonautonomous Poincar\'e oscillator as an example. We demonstrate that these
types differ in their transient dynamics towards a driven steady state and
according to their response to external perturbations. Various possible
realizations of chronotaxic systems are discussed, including systems with
temporal chronotaxicity and interacting chronotaxic systems.Comment: 9 pages, 8 figure
Multi-scale uncertainty quantification in geostatistical seismic inversion
Geostatistical seismic inversion is commonly used to infer the spatial
distribution of the subsurface petro-elastic properties by perturbing the model
parameter space through iterative stochastic sequential
simulations/co-simulations. The spatial uncertainty of the inferred
petro-elastic properties is represented with the updated a posteriori variance
from an ensemble of the simulated realizations. Within this setting, the
large-scale geological (metaparameters) used to generate the petro-elastic
realizations, such as the spatial correlation model and the global a priori
distribution of the properties of interest, are assumed to be known and
stationary for the entire inversion domain. This assumption leads to
underestimation of the uncertainty associated with the inverted models. We
propose a practical framework to quantify uncertainty of the large-scale
geological parameters in seismic inversion. The framework couples
geostatistical seismic inversion with a stochastic adaptive sampling and
Bayesian inference of the metaparameters to provide a more accurate and
realistic prediction of uncertainty not restricted by heavy assumptions on
large-scale geological parameters. The proposed framework is illustrated with
both synthetic and real case studies. The results show the ability retrieve
more reliable acoustic impedance models with a more adequate uncertainty spread
when compared with conventional geostatistical seismic inversion techniques.
The proposed approach separately account for geological uncertainty at
large-scale (metaparameters) and local scale (trace-by-trace inversion)
Fast Stochastic Hierarchical Bayesian MAP for Tomographic Imaging
Any image recovery algorithm attempts to achieve the highest quality
reconstruction in a timely manner. The former can be achieved in several ways,
among which are by incorporating Bayesian priors that exploit natural image
tendencies to cue in on relevant phenomena. The Hierarchical Bayesian MAP
(HB-MAP) is one such approach which is known to produce compelling results
albeit at a substantial computational cost. We look to provide further analysis
and insights into what makes the HB-MAP work. While retaining the proficient
nature of HB-MAP's Type-I estimation, we propose a stochastic
approximation-based approach to Type-II estimation. The resulting algorithm,
fast stochastic HB-MAP (fsHBMAP), takes dramatically fewer operations while
retaining high reconstruction quality. We employ our fsHBMAP scheme towards the
problem of tomographic imaging and demonstrate that fsHBMAP furnishes promising
results when compared to many competing methods.Comment: 5 Pages, 4 Figures, Conference (Accepted to Asilomar 2017
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