8,203 research outputs found
Wavelet-based density estimation for noise reduction in plasma simulations using particles
For given computational resources, the accuracy of plasma simulations using
particles is mainly held back by the noise due to limited statistical sampling
in the reconstruction of the particle distribution function. A method based on
wavelet analysis is proposed and tested to reduce this noise. The method, known
as wavelet based density estimation (WBDE), was previously introduced in the
statistical literature to estimate probability densities given a finite number
of independent measurements. Its novel application to plasma simulations can be
viewed as a natural extension of the finite size particles (FSP) approach, with
the advantage of estimating more accurately distribution functions that have
localized sharp features. The proposed method preserves the moments of the
particle distribution function to a good level of accuracy, has no constraints
on the dimensionality of the system, does not require an a priori selection of
a global smoothing scale, and its able to adapt locally to the smoothness of
the density based on the given discrete particle data. Most importantly, the
computational cost of the denoising stage is of the same order as one time step
of a FSP simulation. The method is compared with a recently proposed proper
orthogonal decomposition based method, and it is tested with three particle
data sets that involve different levels of collisionality and interaction with
external and self-consistent fields
Wavelet-Fourier CORSING techniques for multi-dimensional advection-diffusion-reaction equations
We present and analyze a novel wavelet-Fourier technique for the numerical
treatment of multidimensional advection-diffusion-reaction equations based on
the CORSING (COmpRessed SolvING) paradigm. Combining the Petrov-Galerkin
technique with the compressed sensing approach, the proposed method is able to
approximate the largest coefficients of the solution with respect to a
biorthogonal wavelet basis. Namely, we assemble a compressed discretization
based on randomized subsampling of the Fourier test space and we employ sparse
recovery techniques to approximate the solution to the PDE. In this paper, we
provide the first rigorous recovery error bounds and effective recipes for the
implementation of the CORSING technique in the multi-dimensional setting. Our
theoretical analysis relies on new estimates for the local a-coherence, which
measures interferences between wavelet and Fourier basis functions with respect
to the metric induced by the PDE operator. The stability and robustness of the
proposed scheme is shown by numerical illustrations in the one-, two-, and
three-dimensional case
Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamic properties
The wavelet transform, a family of orthonormal bases, is introduced as a
technique for performing multiresolution analysis in statistical mechanics. The
wavelet transform is a hierarchical technique designed to separate data sets
into sets representing local averages and local differences. Although
one-to-one transformations of data sets are possible, the advantage of the
wavelet transform is as an approximation scheme for the efficient calculation
of thermodynamic and ensemble properties. Even under the most drastic of
approximations, the resulting errors in the values obtained for average
absolute magnetization, free energy, and heat capacity are on the order of 10%,
with a corresponding computational efficiency gain of two orders of magnitude
for a system such as a Ising lattice. In addition, the errors in
the results tend toward zero in the neighborhood of fixed points, as determined
by renormalization group theory.Comment: 13 pages plus 7 figures (PNG
Spin-SILC: CMB polarisation component separation with spin wavelets
We present Spin-SILC, a new foreground component separation method that
accurately extracts the cosmic microwave background (CMB) polarisation and
modes from raw multifrequency Stokes and measurements of the
microwave sky. Spin-SILC is an internal linear combination method that uses
spin wavelets to analyse the spin-2 polarisation signal . The
wavelets are additionally directional (non-axisymmetric). This allows different
morphologies of signals to be separated and therefore the cleaning algorithm is
localised using an additional domain of information. The advantage of spin
wavelets over standard scalar wavelets is to simultaneously and
self-consistently probe scales and directions in the polarisation signal and in the underlying and modes, therefore providing the ability
to perform component separation and - decomposition concurrently for the
first time. We test Spin-SILC on full-mission Planck simulations and data and
show the capacity to correctly recover the underlying cosmological and
modes. We also demonstrate a strong consistency of our CMB maps with those
derived from existing component separation methods. Spin-SILC can be combined
with the pseudo- and pure - spin wavelet estimators presented in a
companion paper to reliably extract the cosmological signal in the presence of
complicated sky cuts and noise. Therefore, it will provide a
computationally-efficient method to accurately extract the CMB and
modes for future polarisation experiments.Comment: 13 pages, 9 figures. Minor changes to match version published in
MNRAS. Map products available at http://www.silc-cmb.org. Companion paper:
arXiv:1605.01414 "Wavelet reconstruction of pure E and B modes for CMB
polarisation and cosmic shear analyses" (B. Leistedt et al.
Intertwining wavelets or Multiresolution analysis on graphs through random forests
We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure
Random Forests and Networks Analysis
D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient
algorithm based on loop-erased random walks to sample uniform spanning trees
and more generally weighted trees or forests spanning a given graph. This
algorithm provides a powerful tool in analyzing structures on networks and
along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we
focused on applications of spanning rooted forests on finite graphs. The
resulting main conclusions are reviewed in this paper by collecting related
theorems, algorithms, heuristics and numerical experiments. A first
foundational part on determinantal structures and efficient sampling procedures
is followed by four main applications: 1) a random-walk-based notion of
well-distributed points in a graph 2) how to describe metastable dynamics in
finite settings by means of Markov intertwining dualities 3) coarse graining
schemes for networks and associated processes 4) wavelets-like pyramidal
algorithms for graph signals.Comment: Survey pape
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