逐次坐标搜索与分量-分量构造的秩1格点规则
Successive Coordinate Search and Component-by-Component Construction of Rank-1 Lattice Rules
AI总结：
本文提出一种基于逐次坐标搜索的改进算法，用于构造秩1格点规则的生成向量，实现更优的最坏误差界，并通过数值实验验证其在权重衰减较慢时的优越性。
摘要：
(fast) component-by-component (CBC)算法是构造加权再生核希尔伯特空间中秩1格点规则生成向量的有效工具。我们考虑乘积权重，为每个维度分配权重。这些权重编码了某个变量（或通过个体权重的乘积变量组）的影响。较小的权重表示较低的重要性。Kuo (2003)证明CBC算法在相应函数空间中达到最优收敛率，但这并不意味着算法会找到具有最小最坏误差的生成向量。事实上并非如此。我们研究了一种分量-分量构造的推广方法，允许基于初始生成向量进行一般性的逐次坐标搜索（SCS），旨在更接近最小最坏误差。所提出的方法允许与CBC算法相同的最坏误差界类型，与初始向量的选择无关。在与[Kuo,2003]相同的权重可和条件下，算法的误差界可以独立于维度d，并在[Kuo,2003]的函数空间中达到相同的最优收敛阶。此外，基于Nuyens和Cools的快速CBC算法的快速版本可用，将算法的计算成本降低到O(d n log(n))次操作，其中n表示函数评估次数。通过以Korobov型生成向量为种子的数值实验表明，新的SCS算法会比CBC算法找到更好的选择，并且当权重衰减较慢时效果更佳。
英文摘要：
The (fast) component-by-component (CBC) algorithm is an efficient tool for the construction of generating vectors for quasi-Monte Carlo rank-1 lattice rules in weighted reproducing kernel Hilbert spaces. We consider product weights, which assigns a weight to each dimension. These weights encode the effect a certain variable (or a group of variables by the product of the individual weights) has. Smaller weights indicate less importance. Kuo (2003) proved that the CBC algorithm achieves the optimal rate of convergence in the respective function spaces, but this does not imply the algorithm will find the generating vector with the smallest worst-case error. In fact it does not. We investigate a generalization of the component-by-component construction that allows for a general successive coordinate search (SCS), based on an initial generating vector, and with the aim of getting closer to the smallest worst-case error. The proposed method admits the same type of worst-case error bounds as the CBC algorithm, independent of the choice of the initial vector. Under the same summability conditions on the weights as in [Kuo,2003] the error bound of the algorithm can be made independent of the dimension $d$ and we achieve the same optimal order of convergence for the function spaces from [Kuo,2003]. Moreover, a fast version of our method, based on the fast CBC algorithm by Nuyens and Cools, is available, reducing the computational cost of the algorithm to $O(d \, n \log(n))$ operations, where $n$ denotes the number of function evaluations. Numerical experiments seeded by a Korobov-type generating vector show that the new SCS algorithm will find better choices than the CBC algorithm and the effect is better when the weights decay slower.
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