Quasi-Monte Carlo integration over Rs based on digital nets

This paper discusses p-weighted integration of functions over the 3-dimensional Euclidean space using quasi-Monte Carlo (QMC) rules combined with an inversion method, where the grobability density function (PDF) is of product form, i.e., a product of uni-variate PDFs for each coordinate in (1....s).<br /> The space of integrands is specified by means of fay-weighted p-norm, p >= 1, which involves coordinate weights y, the partial derivatives of order up to one of the integrands as well as additional weight functions y, and the PDFs 4. The coordinate weights y model the importance of different coordinates or groups of coordinates in the sense of Sloan and Wo & zacute;niakowski, and the weight functions y, are additional parameters of the space which describe the decay of the partial derivatives of the integrands. Fast decaying weights w(x) for x 100 enlarge the space. of functions with finite norm, but decrease the convergence rate of the worst-case epror of the proposed algorithms.<br /> Our algorithms for integration use digitally shifted digital nets in combination with an inversion method. We study the (root) mean squared worst-case error with respect to random digital shifts. The obtained error bounds depend on the choice of weight functions, and coordinate weights y. Under certain conditions on y, these bounds hold uniformly for all dimensions s.<br /> Numerical experiments demonstrate the effectiveness of the proposed algorithms.