论文标题
关于集成分析功能的晶格规则的注释
A note on Korobov lattice rules for integration of analytic functions
论文作者
论文摘要
我们研究了分析周期函数的加权Korobov空间的数值整合，傅立叶系数的衰减呈指数速度。特别是，我们对错误如何取决于尺寸$ d $感兴趣。许多最近的论文涉及此问题或类似问题，并为各种障碍概念提供了匹配的必要条件。在大多数情况下，即使是简单的算法，也已经知道可以实现这些障碍的概念。但是，文献中存在一个差距：对于指数型障碍性的概念，人们知道匹配必要和充分的条件，但到目前为止，尚无明确算法得出所需的结果。 在本文中，我们缩小了这一差距，并证明了Korobov晶格规则是合适的算法，以实现指数型拖延性，以在分析周期性功能的加权Korobov空间中集成。
We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension $d$. Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result. In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions.
