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Sparse grid quadrature on products of spheres
We examine sparse grid quadrature on weighted tensor products (WTP) of
reproducing kernel Hilbert spaces on products of the unit sphere, in the case
of worst case quadrature error for rules with arbitrary quadrature weights. We
describe a dimension adaptive quadrature algorithm based on an algorithm of
Hegland (2003), and also formulate a version of Wasilkowski and Wozniakowski's
WTP algorithm (1999), here called the WW algorithm. We prove that the dimension
adaptive algorithm is optimal in the sense of Dantzig (1957) and therefore no
greater in cost than the WW algorithm. Both algorithms therefore have the
optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and
Wozniakowski (1999). A numerical example shows that, even though the asymptotic
convergence rate is optimal, if the dimension weights decay slowly enough, and
the dimensionality of the problem is large enough, the initial convergence of
the dimension adaptive algorithm can be slow.Comment: 34 pages, 6 figures. Accepted 7 January 2015 for publication in
Numerical Algorithms. Revised at page proof stage to (1) update email
address; (2) correct the accent on "Wozniakowski" on p. 7; (3) update
reference 2; (4) correct references 3, 18 and 2
Comparison of some Reduced Representation Approximations
In the field of numerical approximation, specialists considering highly
complex problems have recently proposed various ways to simplify their
underlying problems. In this field, depending on the problem they were tackling
and the community that are at work, different approaches have been developed
with some success and have even gained some maturity, the applications can now
be applied to information analysis or for numerical simulation of PDE's. At
this point, a crossed analysis and effort for understanding the similarities
and the differences between these approaches that found their starting points
in different backgrounds is of interest. It is the purpose of this paper to
contribute to this effort by comparing some constructive reduced
representations of complex functions. We present here in full details the
Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM)
together with other approaches that enter in the same category
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