Component-by-component construction of shifted Halton sequences

Abstract: We study quasi-Monte Carlo integration in a weighted anchored Sobolev space. As the underlying integration nodes we consider Halton sequences in prime bases $\boldsymbol{p}=(p_1,\ldots,p_s)$ which are shifted with a $\boldsymbol{p}$-adic shift based on $\boldsymbol{p}$-adic arithmetic. The error is studied in the worst-case setting. In a paper, Hellekalek together with the authors of this article proved optimal error bounds in the root mean square sense, where the mean was extended over the uncountable set of all possible $\boldsymbol{p}$-adic shifts. Here we show that candidates for good shifts can in fact be chosen from a finite set and can be found by a component-by-component algorithm.