用简化的快速逐次坐标搜索算法构造数值积分格点
Constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm
摘要：
本文研究了在加权Korobov、Walsh和索伯列夫空间上构造高质量拟蒙特卡罗方法积分规则节点集的有效算法。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值衰减足够快的情况。新的SCS算法设计用于格点的构造，并在一个改进的版本中用于多项式格点的构造，相应的积分规则可以用于处理不同类型的函数空间中的函数。我们证明了由我们的算法构造的积分规则满足最优收敛阶的误差界。此外，我们给出了详细的高效实现，使我们获得了相当大的速度提高以前已知的SCS算法。数值结果说明了这种改进。我们的结果所获得的加速可能是特别感兴趣的上下文中的QMC随机系数的偏微分方程，其中的尺寸和所需的numberof点通常是非常大的。此外，我们的主要定理产生以前未知的推广早期的结果。
英文摘要：
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.
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