In this paper, we study an efficient algorithm for constructing point sets underlying quasi-Monte Carlo integration rules for weighted Korobov classes. The algorithm presented is a reduced fast component-by-component digit-by-digit (CBC-DBD) algorithm, which useful for to situations where the weights in the function space show a sufficiently fast decay. The advantage of the algorithm presented here is that the computational effort can be independent of the dimension of the integration problem to be treated if suitable assumptions on the integrand are met. The new reduced CBC-DBD algorithm is designed to work for the construction of lattice point sets, and the corresponding integration rules (so-called lattice rules) can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithm satisfy error bounds of almost optimal convergence order. Furthermore, we give details on an efficient implementation such that we obtain a considerable speed-up of a previously known CBC-DBD algorithm that has been studied before. This improvement is illustrated by numerical results.
翻译:在本文中,我们研究了一种高效的算法,用于为加权的Korobov 类类构建点设置准蒙特卡洛整合规则的基础。 显示的算法是一种速成件逐成数字数字( CBC- DBD)算法,对于功能空间的重量显示足够快速衰减的情况有用。 这里所介绍的算法的优点是,计算努力可以独立于一体化问题的层面,如果满足了对指数的恰当假设,则需要加以处理。 新的CBC- DBD算法旨在为建造拉特基点集而工作,相应的集成规则(所谓的拉特克规则)可以用来处理不同功能空间的功能。 我们表明,我们的算法构建的集成规则可以满足几乎最佳的汇合顺序的错误界限。 此外,我们详细介绍了高效率的实施,这样我们就能大大加速以前研究过的CBC- DBD算法。 这一改进可以用数字结果来说明。