Title: Weighted compound integration rules with higher order convergence for all N
Authors: Hickernell, Fred J.
Kritzer, Peter
Kuo, Frances Y.
Nuyens, Dirk
Issue Date: Aug-2009
Publisher: Department of Mathematics and Statistics, University of New South Wales
Series Title: Applied Mathematics Reports vol:AMR09/19
Abstract: Quasi-Monte Carlo integration rules, which are equal-weight sample
averages of function values, have been popular for approximating
multivariate integrals due to their superior convergence rate of order
close to $1/N$ or better, compared to the order $1/\sqrt{N}$ of simple
Monte Carlo algorithms. For practical applications, it is desirable to be
able to increase the total number of sampling points $N$ one at a time
until a desired accuracy is met, while keeping all existing evaluations.
We show that, unfortunately, it is impossible to get better than order
$1/N$ convergence for all values of $N$ by adding equally-weighted
sampling points in this manner. On the other hand, we prove that a
convergence of order $N^{-\alpha}$ for $\alpha>1$ can be achieved by
weighting the sampling points, that is, by using a weighted compound
integration rule. We apply our theory to lattice sequences and present
some numerical results. Our theory also applies to digital sequences.
Publication status: published
KU Leuven publication type: IR
Appears in Collections:Numerical Analysis and Applied Mathematics Section

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