On the component by component construction of polynomial lattice point sets for numerical integration in weighted Sobolev spaces

Polynomial lattice point sets are polynomial versions of classical lattice point sets and among the most widely used classes of node sets for quasi-Monte Carlo integration. In this paper, we study the worst-case integration error of digitally shifted polynomial lattice point sets and give step by step construction algorithms to obtain polynomial lattices that achieve a low worst-case error in certain weighted Sobolev spaces. The construction algorithm is a so-called component by component algorithm, choosing one component of the relevant point set at a time. Furthermore, under certain conditions on the weights, we achieve that there is only a polynomial or even no dependence of the worst-case error on the dimension of the integration problem.