Liberating the weights

Abstract

A partial answer to why quasi-Monte Carlo (QMC) algorithms work well for multivariate integration was given in Sloan and Woźniakowski (J. Complexity 14 (1998) 1–33) by introducing weighted spaces. In these spaces the importance of successive coordinate directions is quantified by a sequence of weights. However, to be able to make use of weighted spaces for a particular application one has to make a choice of the weights. In this work, we take a more general view of the weights by allowing them to depend arbitrarily not only on the coordinates but also on the number of variables. Liberating the weights in this way allows us to give a recommendation for how to choose the weights in practice. This recommendation results from choosing the weights so as to minimize the error bound. We also consider how best to choose the underlying weighted Sobolev space within which to carry out the analysis. We revisit also lower bounds on the worst-case error, which change in many minor ways now, since the weights are allowed to depend on the number of variables, and we do not assume that the weights are uniformly bounded as has been assumed in previous papers. Necessary and sufficient conditions for QMC tractability and strong QMC tractability are obtained for the weighted Sobolev spaces with general weights. In the final section, we show that the analysis of variance decomposition of functions from one of the Sobolev spaces is equivalent to the decomposition of functions with respect to an orthogonal decomposition of this space.