In this paper, we study an efficient algorithm for constructing node sets of high-quality quasi-Monte Carlo integration rules for weighted Korobov, Walsh, and Sobolev spaces. The algorithm presented is a reduced fast successive coordinate search (SCS) algorithm, which is adapted to situations where the weights in the function space show a sufficiently fast decay. The new SCS algorithm is designed to work for the construction of lattice points, and, in a modified version, for polynomial lattice points, and the corresponding integration rules can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithms satisfy error bounds of optimal convergence order. Furthermore, we give details on efficient implementation such that we obtain a considerable speed-up of previously known SCS algorithms. This improvement is illustrated by numerical results. The speed-up obtained by our results may be of particular interest in the context of QMC for PDEs with random coefficients, where both the dimension and the required numberof points are usually very large. Furthermore, our main theorems yield previously unknown generalizations of earlier results.

中文翻译：

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用简化的快速连续坐标搜索算法构建数值积分的格点

在本文中，我们研究了一种有效的算法，用于构建加权 Korobov、Walsh 和 Sobolev 空间的高质量准蒙特卡罗积分规则的节点集。所提出的算法是一种简化的快速连续坐标搜索 (SCS) 算法，它适用于函数空间中的权重显示出足够快的衰减的情况。新的 SCS 算法设计用于构建格点，在修改版本中用于多项式格点，相应的积分规则可用于处理不同类型函数空间中的函数。我们表明，由我们的算法构建的积分规则满足最佳收敛阶数的误差界限。此外，我们提供了有关有效实现的详细信息，以便我们获得先前已知的 SCS 算法的显着加速。数值结果说明了这种改进。我们的结果获得的加速可能在具有随机系数的 PDE 的 QMC 上下文中特别有趣，其中维度和所需的点数通常都非常大。此外，我们的主要定理产生了先前结果的先前未知的概括。