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We study quasi-Monte Carlo (QMC) methods for numerical integration ofmultivariate functions defined over the high-dimensional unit cube. Latticerules and polynomial lattice rules, which are special classes of QMC methods,have been intensively studied and the so-called component-by-component (CBC)algorithm has been well-established to construct rules which achieve the almostoptimal rate of convergence with good tractability properties for givensmoothness and set of weights. Since the CBC algorithm constructs rules forgiven smoothness and weights, not much is known when such rules are used forfunction classes with different smoothness and/or weights.  In this paper we prove that a lattice rule constructed by the CBC algorithmfor the weighted Korobov space with given smoothness and weights achieves thealmost optimal rate of convergence with good tractability properties forgeneral classes of smoothness and weights which satisfy some summabilityconditions. Such a stability result also can be shown for polynomial latticerules in weighted Walsh spaces. We further give bounds on the weighted stardiscrepancy and discuss the tractability properties for these QMC rules. Theresults are comparable to those obtained for Halton, Sobol and Niederreitersequences.

Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm