Title: Weighted compound integration rules with higher order convergence for all N Authors: Hickernell, Fred J.Kritzer, PeterKuo, 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. URI: http://www.maths.unsw.edu.au/applied/pubs/apppreprints2009.html Publication status: published KU Leuven publication type: IR Appears in Collections: NUMA, Numerical Analysis and Applied Mathematics Section