2,651 research outputs found
The rational SPDE approach for Gaussian random fields with general smoothness
A popular approach for modeling and inference in spatial statistics is to
represent Gaussian random fields as solutions to stochastic partial
differential equations (SPDEs) of the form , where
is Gaussian white noise, is a second-order differential
operator, and is a parameter that determines the smoothness of .
However, this approach has been limited to the case ,
which excludes several important models and makes it necessary to keep
fixed during inference.
We propose a new method, the rational SPDE approach, which in spatial
dimension is applicable for any , and thus remedies
the mentioned limitation. The presented scheme combines a finite element
discretization with a rational approximation of the function to
approximate . For the resulting approximation, an explicit rate of
convergence to in mean-square sense is derived. Furthermore, we show that
our method has the same computational benefits as in the restricted case
. Several numerical experiments and a statistical
application are used to illustrate the accuracy of the method, and to show that
it facilitates likelihood-based inference for all model parameters including
.Comment: 28 pages, 4 figure
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Stability and covergence properties of Bergman Kernel methods for numerical conformal mapping
In this paper we study the stability and convergence properties of Bergman kernel methods, for the numerical conform al mapping of simply and doubly- connected domains. In particular, by using certain well-known results of Carleman, we establish a characterization of the level of instability in the methods, in terms of the geometry of the domain under consideration. We also explain how certain known convergence results can provide some theoretical justification of the observed improvement in accuracy which is achieved by the methods, when the basis set used contains functions that reflect the main singular behaviour of the conformal map
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
Discontinuities in numerical radiative transfer
Observations and magnetohydrodynamic simulations of solar and stellar
atmospheres reveal an intermittent behavior or steep gradients in physical
parameters, such as magnetic field, temperature, and bulk velocities. The
numerical solution of the stationary radiative transfer equation is
particularly challenging in such situations, because standard numerical methods
may perform very inefficiently in the absence of local smoothness. However, a
rigorous investigation of the numerical treatment of the radiative transfer
equation in discontinuous media is still lacking. The aim of this work is to
expose the limitations of standard convergence analyses for this problem and to
identify the relevant issues. Moreover, specific numerical tests are performed.
These show that discontinuities in the atmospheric physical parameters
effectively induce first-order discontinuities in the radiative transfer
equation, reducing the accuracy of the solution and thwarting high-order
convergence. In addition, a survey of the existing numerical schemes for
discontinuous ordinary differential systems and interpolation techniques for
discontinuous discrete data is given, evaluating their applicability to the
radiative transfer problem
Math Active Learning Lab: Math 103 College Algebra Notebook
This course notebook has been designed for students of Math 103 (College Algebra) at the University of North Dakota. It has been designed to help you get the most out of the ALEKS resources and your time. Topics in the Notebook are organized by weekly learning module. Space for notes from ALEKS learning pages, e-book and videos directs you to essential concepts. Examples and âYou Try Itâ problems have been carefully chosen to help you focus on these essential concepts. Completed Notebook is an invaluable tool when studying for exams.https://commons.und.edu/oers/1024/thumbnail.jp
Efficient contact determination between geometric models
http://archive.org/details/efficientcontact00linmN
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