4,825 research outputs found
Smoothed Particle Hydrodynamics and Magnetohydrodynamics
This paper presents an overview and introduction to Smoothed Particle
Hydrodynamics and Magnetohydrodynamics in theory and in practice. Firstly, we
give a basic grounding in the fundamentals of SPH, showing how the equations of
motion and energy can be self-consistently derived from the density estimate.
We then show how to interpret these equations using the basic SPH interpolation
formulae and highlight the subtle difference in approach between SPH and other
particle methods. In doing so, we also critique several `urban myths' regarding
SPH, in particular the idea that one can simply increase the `neighbour number'
more slowly than the total number of particles in order to obtain convergence.
We also discuss the origin of numerical instabilities such as the pairing and
tensile instabilities. Finally, we give practical advice on how to resolve
three of the main issues with SPMHD: removing the tensile instability,
formulating dissipative terms for MHD shocks and enforcing the divergence
constraint on the particles, and we give the current status of developments in
this area. Accompanying the paper is the first public release of the NDSPMHD
SPH code, a 1, 2 and 3 dimensional code designed as a testbed for SPH/SPMHD
algorithms that can be used to test many of the ideas and used to run all of
the numerical examples contained in the paper.Comment: 44 pages, 14 figures, accepted to special edition of J. Comp. Phys.
on "Computational Plasma Physics". The ndspmhd code is available for download
from http://users.monash.edu.au/~dprice/ndspmhd
Interpolating point spread function anisotropy
Planned wide-field weak lensing surveys are expected to reduce the
statistical errors on the shear field to unprecedented levels. In contrast,
systematic errors like those induced by the convolution with the point spread
function (PSF) will not benefit from that scaling effect and will require very
accurate modeling and correction. While numerous methods have been devised to
carry out the PSF correction itself, modeling of the PSF shape and its spatial
variations across the instrument field of view has, so far, attracted much less
attention. This step is nevertheless crucial because the PSF is only known at
star positions while the correction has to be performed at any position on the
sky. A reliable interpolation scheme is therefore mandatory and a popular
approach has been to use low-order bivariate polynomials. In the present paper,
we evaluate four other classical spatial interpolation methods based on splines
(B-splines), inverse distance weighting (IDW), radial basis functions (RBF) and
ordinary Kriging (OK). These methods are tested on the Star-challenge part of
the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) simulated data and
are compared with the classical polynomial fitting (Polyfit). We also test all
our interpolation methods independently of the way the PSF is modeled, by
interpolating the GREAT10 star fields themselves (i.e., the PSF parameters are
known exactly at star positions). We find in that case RBF to be the clear
winner, closely followed by the other local methods, IDW and OK. The global
methods, Polyfit and B-splines, are largely behind, especially in fields with
(ground-based) turbulent PSFs. In fields with non-turbulent PSFs, all
interpolators reach a variance on PSF systematics better than
the upper bound expected by future space-based surveys, with
the local interpolators performing better than the global ones
Fast space-variant elliptical filtering using box splines
The efficient realization of linear space-variant (non-convolution) filters
is a challenging computational problem in image processing. In this paper, we
demonstrate that it is possible to filter an image with a Gaussian-like
elliptic window of varying size, elongation and orientation using a fixed
number of computations per pixel. The associated algorithm, which is based on a
family of smooth compactly supported piecewise polynomials, the
radially-uniform box splines, is realized using pre-integration and local
finite-differences. The radially-uniform box splines are constructed through
the repeated convolution of a fixed number of box distributions, which have
been suitably scaled and distributed radially in an uniform fashion. The
attractive features of these box splines are their asymptotic behavior, their
simple covariance structure, and their quasi-separability. They converge to
Gaussians with the increase of their order, and are used to approximate
anisotropic Gaussians of varying covariance simply by controlling the scales of
the constituent box distributions. Based on the second feature, we develop a
technique for continuously controlling the size, elongation and orientation of
these Gaussian-like functions. Finally, the quasi-separable structure, along
with a certain scaling property of box distributions, is used to efficiently
realize the associated space-variant elliptical filtering, which requires O(1)
computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201
Non-equispaced B-spline wavelets
This paper has three main contributions. The first is the construction of
wavelet transforms from B-spline scaling functions defined on a grid of
non-equispaced knots. The new construction extends the equispaced,
biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new
construction is based on the factorisation of wavelet transforms into lifting
steps. The second and third contributions are new insights on how to use these
and other wavelets in statistical applications. The second contribution is
related to the bias of a wavelet representation. It is investigated how the
fine scaling coefficients should be derived from the observations. In the
context of equispaced data, it is common practice to simply take the
observations as fine scale coefficients. It is argued in this paper that this
is not acceptable for non-interpolating wavelets on non-equidistant data.
Finally, the third contribution is the study of the variance in a
non-orthogonal wavelet transform in a new framework, replacing the numerical
condition as a measure for non-orthogonality. By controlling the variances of
the reconstruction from the wavelet coefficients, the new framework allows us
to design wavelet transforms on irregular point sets with a focus on their use
for smoothing or other applications in statistics.Comment: 42 pages, 2 figure
Penalized Likelihood and Bayesian Function Selection in Regression Models
Challenging research in various fields has driven a wide range of
methodological advances in variable selection for regression models with
high-dimensional predictors. In comparison, selection of nonlinear functions in
models with additive predictors has been considered only more recently. Several
competing suggestions have been developed at about the same time and often do
not refer to each other. This article provides a state-of-the-art review on
function selection, focusing on penalized likelihood and Bayesian concepts,
relating various approaches to each other in a unified framework. In an
empirical comparison, also including boosting, we evaluate several methods
through applications to simulated and real data, thereby providing some
guidance on their performance in practice
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