1,604 research outputs found
A Practical Box Spline Compendium
Box splines provide smooth spline spaces as shifts of a single generating
function on a lattice and so generalize tensor-product splines. Their elegant
theory is laid out in classical papers and a summarizing book. This compendium
aims to succinctly but exhaustively survey symmetric low-degree box splines
with special focus on two and three variables. Tables contrast the lattices,
supports, analytic and reconstruction properties, and list available
implementations and code.Comment: 15 pages, 10 figures, 8 table
Near-best quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains
In this paper, we present new quasi-interpolating spline schemes defined on
3D bounded domains, based on trivariate quartic box splines on type-6
tetrahedral partitions and with approximation order four. Such methods can be
used for the reconstruction of gridded volume data. More precisely, we propose
near-best quasi-interpolants, i.e. with coefficient functionals obtained by
imposing the exactness of the quasi-interpolants on the space of polynomials of
total degree three and minimizing an upper bound for their infinity norm. In
case of bounded domains the main problem consists in the construction of the
coefficient functionals associated with boundary generators (i.e. generators
with supports not completely inside the domain), so that the functionals
involve data points inside or on the boundary of the domain.
We give norm and error estimates and we present some numerical tests,
illustrating the approximation properties of the proposed quasi-interpolants,
and comparisons with other known spline methods. Some applications with real
world volume data are also provided.Comment: In the new version of the paper, we have done some minor revisions
with respect to the previous version, CALCOLO, Published online: 10 October
201
Sampling from a system-theoretic viewpoint
This paper studies a system-theoretic approach to the problem of reconstructing an analog signal from its samples. The idea, borrowed from earlier treatments in the control literature, is to address the problem as a hybrid model-matching problem in which performance is measured by system norms. \ud
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The paper is split into three parts. In Part I we present the paradigm and revise the lifting technique, which is our main technical tool. In Part II optimal samplers and holds are designed for various analog signal reconstruction problems. In some cases one component is fixed while the remaining are designed, in other cases all three components are designed simultaneously. No causality requirements are imposed in Part II, which allows to use frequency domain arguments, in particular the lifted frequency response as introduced in Part I. In Part III the main emphasis is placed on a systematic incorporation of causality constraints into the optimal design of reconstructors. We consider reconstruction problems, in which the sampling (acquisition) device is given and the performance is measured by the -norm of the reconstruction error. The problem is solved under the constraint that the optimal reconstructor is -causal for a given i.e., that its impulse response is zero in the time interval where is the sampling period. We derive a closed-form state-space solution of the problem, which is based on the spectral factorization of a rational transfer function
Symmetric box-splines on root lattices
AbstractRoot lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions, due to their highly symmetric structure. One root lattice, the Cartesian grid, is almost exclusively used since it matches the coordinate grid; but it is less efficient than other root lattices. Box-splines, on the other hand, generalize tensor-product B-splines by allowing non-Cartesian directions. They provide, in any number of dimensions, higher-order reconstructions of fields, often of higher efficiency than tensored B-splines. But on non-Cartesian lattices, such as the BCC (Body-Centered Cubic) or the FCC (Face-Centered Cubic) lattice, only some box-splines and then only up to dimension three have been investigated.This paper derives and completely characterizes efficient symmetric box-spline reconstruction filters on all irreducible root lattices that exist in any number of dimensions n≥2 (n≥3 for Dn and Dn∗ lattices). In all cases, box-splines are constructed by convolution using the lattice directions, generalizing the known constructions in two and three variables. For each box-spline, we document the basic properties for computational use: the polynomial degree, the continuity, the linear independence of shifts on the lattice and optimal quasi-interpolants for fast approximation of fields
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
Boosting Functional Response Models for Location, Scale and Shape with an Application to Bacterial Competition
We extend Generalized Additive Models for Location, Scale, and Shape (GAMLSS)
to regression with functional response. This allows us to simultaneously model
point-wise mean curves, variances and other distributional parameters of the
response in dependence of various scalar and functional covariate effects. In
addition, the scope of distributions is extended beyond exponential families.
The model is fitted via gradient boosting, which offers inherent model
selection and is shown to be suitable for both complex model structures and
highly auto-correlated response curves. This enables us to analyze bacterial
growth in \textit{Escherichia coli} in a complex interaction scenario,
fruitfully extending usual growth models.Comment: bootstrap confidence interval type uncertainty bounds added; minor
changes in formulation
Probabilistic Line Searches for Stochastic Optimization
In deterministic optimization, line searches are a standard tool ensuring
stability and efficiency. Where only stochastic gradients are available, no
direct equivalent has so far been formulated, because uncertain gradients do
not allow for a strict sequence of decisions collapsing the search space. We
construct a probabilistic line search by combining the structure of existing
deterministic methods with notions from Bayesian optimization. Our method
retains a Gaussian process surrogate of the univariate optimization objective,
and uses a probabilistic belief over the Wolfe conditions to monitor the
descent. The algorithm has very low computational cost, and no user-controlled
parameters. Experiments show that it effectively removes the need to define a
learning rate for stochastic gradient descent.Comment: Extended version of the NIPS '15 conference paper, includes detailed
pseudo-code, 59 pages, 35 figure
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