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In this paper, we study quasi-Monte Carlo (QMC) integration in weightedSobolev spaces. In contrast to many previous results the QMC algorithmsconsidered here are of open type, i.e., they are extensible in the number ofsample points without having to discard the samples already used. As theunderlying integration nodes we consider randomized Halton sequences in primebases $\boldsymbol{p}=(p_1,...,p_s)$ for which we study the root mean square(RMS) worst-case error. The randomization method is a $\boldsymbol{p}$-adicshift which is based on $\boldsymbol{p}$-adic arithmetic.  The obtained error bounds are optimal in the order of magnitude of the numberof sample nodes. Furthermore we obtain conditions on the coordinate weightsunder which the error bounds are independent of the dimension $s$. In terms ofthe field of Information-Based Complexity this means that the corresponding QMCrule achieves a strong polynomial tractability error bound. Our findings on theRMS worst-case error of randomized Halton sequences can be carried over to theRMS $L_2$-discrepancy.  Except for the $\boldsymbol{p}$-adic shift our results are fully constructiveand no search algorithms (such as the component-by-component algorithm) arerequired.

Open type quasi-Monte Carlo integration based on Halton sequences in weighted Sobolev spaces