We study quasi-Monte Carlo (QMC) methods for numerical integration of multivariate functions defined over the high-dimensional unit cube. Lattice rules 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 almost optimal rate of convergence with good tractability properties for given smoothness and set of weights. Since the CBC algorithm constructs rules for given smoothness and weights, not much is known when such rules are used for function classes with different smoothness and/or weights.

In this paper we prove that a lattice rule constructed by the CBC algorithm for the weighted Korobov space with given smoothness and weights achieves the almost optimal rate of convergence with good tractability properties for general classes of smoothness and weights which satisfy some summability conditions. Such a stability result also can be shown for polynomial lattice rules in weighted Walsh spaces. We further give bounds on the weighted star discrepancy and discuss the tractability properties for these QMC rules. The results are comparable to those obtained for Halton, Sobol and Niederreiter sequences.

我们研究在高维单位立方体上定义的多元函数的数值积分的拟蒙特卡罗（QMC）方法。已经对作为QMC方法特殊类的格规则和多项式格规则进行了深入的研究，并且已经很好地建立了所谓的逐分量（CBC）算法，以构造可实现几乎最佳收敛速度的规则。对于给定的平滑度和权重集，具有良好的可牵引性。由于CBC算法为给定的平滑度和权重构造规则，因此将此类规则用于具有不同平滑度和/或权重的函数类时了解不多。

在本文中，我们证明了由CBC算法构造的具有给定平滑度和权重的加权Korobov空间的晶格规则，对于满足某些求和条件的一般类别的平滑度和权重，可以实现几乎最佳的收敛速度，并具有良好的可扩展性。对于加权沃尔什空间中的多项式格规则，也可以显示这种稳定性结果。我们进一步给出了加权星号差异的界限，并讨论了这些QMC规则的可延展性。结果与Halton，Sobol和Niederreiter序列获得的结果相当。