1,385 research outputs found
Canonical sets of best L1-approximation
In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L 1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities
Higher order mesh curving using geometry curvature extrapolation
A higher order mesh curving method is developed which uses information from the geometry to determine the appropriate curvature of edges in the interior of the mesh. Edges are represented using four point Bézier curves to determine the positions of higher order edge points. Higher order face and volume points are positioned using the basis functions for serendipity face and volume elements. Parameters are defined which allow user specified control over element quality and the propagation of curvature in the mesh. Curved higher order meshes are shown for test cases in both two and three dimensions
Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws
We present a new perspective on the use of weighted essentially
nonoscillatory (WENO) reconstructions in high-order methods for scalar
hyperbolic conservation laws. The main focus of this work is on nonlinear
stabilization of continuous Galerkin (CG) approximations. The proposed
methodology also provides an interesting alternative to WENO-based limiters for
discontinuous Galerkin (DG) methods. Unlike Runge--Kutta DG schemes that
overwrite finite element solutions with WENO reconstructions, our approach uses
a reconstruction-based smoothness sensor to blend the numerical viscosity
operators of high- and low-order stabilization terms. The so-defined WENO
approximation introduces low-order nonlinear diffusion in the vicinity of
shocks, while preserving the high-order accuracy of a linearly stable baseline
discretization in regions where the exact solution is sufficiently smooth. The
underlying reconstruction procedure performs Hermite interpolation on stencils
consisting of a mesh cell and its neighbors. The amount of numerical
dissipation depends on the relative differences between partial derivatives of
reconstructed candidate polynomials and those of the underlying finite element
approximation. All derivatives are taken into account by the employed
smoothness sensor. To assess the accuracy of our CG-WENO scheme, we derive
error estimates and perform numerical experiments. In particular, we prove that
the consistency error of the nonlinear stabilization is of the order ,
where is the polynomial degree. This estimate is optimal for general
meshes. For uniform meshes and smooth exact solutions, the experimentally
observed rate of convergence is as high as
Efficient Bayesian-based Multi-View Deconvolution
Light sheet fluorescence microscopy is able to image large specimen with high
resolution by imaging the sam- ples from multiple angles. Multi-view
deconvolution can significantly improve the resolution and contrast of the
images, but its application has been limited due to the large size of the
datasets. Here we present a Bayesian- based derivation of multi-view
deconvolution that drastically improves the convergence time and provide a fast
implementation utilizing graphics hardware.Comment: 48 pages, 20 figures, 1 table, under review at Nature Method
Trends in Mathematical Imaging and Surface Processing
Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments
SATMC: Spectral Energy Distribution Analysis Through Markov Chains
We present the general purpose spectral energy distribution (SED) fitting
tool SED Analysis Through Markov Chains (SATMC). Utilizing Monte Carlo Markov
Chain (MCMC) algorithms, SATMC fits an observed SED to SED templates or models
of the user's choice to infer intrinsic parameters, generate confidence levels
and produce the posterior parameter distribution. Here we describe the key
features of SATMC from the underlying MCMC engine to specific features for
handling SED fitting. We detail several test cases of SATMC, comparing results
obtained to traditional least-squares methods, which highlight its accuracy,
robustness and wide range of possible applications. We also present a sample of
submillimetre galaxies that have been fitted using the SED synthesis routine
GRASIL as input. In general, these SMGs are shown to occupy a large volume of
parameter space, particularly in regards to their star formation rates which
range from ~30-3000 M_sun yr^-1 and stellar masses which range from
~10^10-10^12 M_sun. Taking advantage of the Bayesian formalism inherent to
SATMC, we also show how the fitting results may change under different
parametrizations (i.e., different initial mass functions) and through
additional or improved photometry, the latter being crucial to the study of
high-redshift galaxies.Comment: 17 pages, 11 figures, MNRAS accepte
GRID2D/3D: A computer program for generating grid systems in complex-shaped two- and three-dimensional spatial domains. Part 1: Theory and method
An efficient computer program, called GRID2D/3D was developed to generate single and composite grid systems within geometrically complex two- and three-dimensional (2- and 3-D) spatial domains that can deform with time. GRID2D/3D generates single grid systems by using algebraic grid generation methods based on transfinite interpolation in which the distribution of grid points within the spatial domain is controlled by stretching functions. All single grid systems generated by GRID2D/3D can have grid lines that are continuous and differentiable everywhere up to the second-order. Also, grid lines can intersect boundaries of the spatial domain orthogonally. GRID2D/3D generates composite grid systems by patching together two or more single grid systems. The patching can be discontinuous or continuous. For continuous composite grid systems, the grid lines are continuous and differentiable everywhere up to the second-order except at interfaces where different single grid systems meet. At interfaces where different single grid systems meet, the grid lines are only differentiable up to the first-order. For 2-D spatial domains, the boundary curves are described by using either cubic or tension spline interpolation. For 3-D spatial domains, the boundary surfaces are described by using either linear Coon's interpolation, bi-hyperbolic spline interpolation, or a new technique referred to as 3-D bi-directional Hermite interpolation. Since grid systems generated by algebraic methods can have grid lines that overlap one another, GRID2D/3D contains a graphics package for evaluating the grid systems generated. With the graphics package, the user can generate grid systems in an interactive manner with the grid generation part of GRID2D/3D. GRID2D/3D is written in FORTRAN 77 and can be run on any IBM PC, XT, or AT compatible computer. In order to use GRID2D/3D on workstations or mainframe computers, some minor modifications must be made in the graphics part of the program; no modifications are needed in the grid generation part of the program. This technical memorandum describes the theory and method used in GRID2D/3D
The MACHO Project LMC Microlensing Results from the First Two Years and the Nature of the Galactic Dark Halo
The MACHO Project is a search for dark matter in the form of massive compact
halo objects (Machos). Photometric monitoring of millions of stars in the Large
Magellanic Cloud (LMC), Small Magellanic Cloud (SMC), and Galactic bulge is
used to search for gravitational microlensing events caused by these otherwise
invisible objects. Analysis of the first 2.1 years of photometry of 8.5 million
stars in the LMC reveals 8 candidate microlensing events. This is substantially
more than the number expected () from lensing by known stellar
populations. The timescales (\that) of the events range from 34 to 145 days.
We estimate the total microlensing optical depth towards the LMC from events
with 2 < \that < 200 days to be \tau_2^{200} = 2.9 ^{+1.4}_{-0.9} \ten{-7}
based upon our 8 event sample. This exceeds the optical depth, \tau_{\rm
backgnd} = 0.5 \ten{-7}, expected from known stars, and the difference is to
be compared with the optical depth predicted for a ``standard" halo composed
entirely of Machos: \tau_{halo} = 4.7\ten{-7}. Likelihood analysis gives a
fairly model independent estimate of the halo mass in Machos within 50 kpc of
2.0^{+1.2}_{-0.7} \ten{11} \msun, about half of the ``standard halo" value.
We also find a most probable Macho mass of 0.5^{+0.3}_{-0.2}\msun, although
this value is strongly model dependent. Additionally, the absence of short
duration events places stringent upper limits on the contribution of low-mass
Machos: objects from 10^{-4} \msun to 0.03 \msun contribute \simlt 20\%
of the ``standard" dark halo.Comment: Latex, 54 pages, uses aas2pp4.sty and astrobib.sty, with 24 out of 26
Postscript figures in gzipped tar file. 2 extra greyscale figures and/or full
paper available from ftp://igpp.llnl.gov/pub/macho/LMC2/ Submitted to ApJ,
June 199
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