论文标题
高维组的非晶格覆盖和Quanitization
Non-lattice covering and quanitization of high dimensional sets
论文作者
论文摘要
本文考虑的主要问题是构建和理论研究$ d $维的$ n $点覆盖物$ [ - 1,1]^d $。 $ d $的目标值在5到50之间； $ n $可以分别为数百或数千，而设计（积分集）是嵌套的。本文是我们论文\ cite {us}的延续，在理论上我们研究了几个简单的方案，并在数字上研究了更多。在本文中，我们扩展了\ cite {us}的理论构造，用于研究设计的设计，这些设计比在\ cite {us}中所研究的设计优于理论上的设计。我们还扩展了与\ cite {us}中的数值相比，我们扩展了新的构造方案的构造（在嵌套设计类中）。鉴于量化问题与覆盖问题的紧密联系，我们将理论近似值和实际建议扩展到了立方体$ [-1,1]^d $中有效量化设计的构建问题。在最后一部分中，我们讨论了$ d $二维单纯形中覆盖和量化的问题；本卷的共同编辑迈克尔·弗拉哈蒂斯（Michael Vrahatis）教授将这个问题的实际意义传达给了作者。
The main problem considered in this paper is construction and theoretical study of efficient $n$-point coverings of a $d$-dimensional cube $[-1,1]^d$. Targeted values of $d$ are between 5 and 50; $n$ can be in hundreds or thousands and the designs (collections of points) are nested. This paper is a continuation of our paper \cite{us}, where we have theoretically investigated several simple schemes and numerically studied many more. In this paper, we extend the theoretical constructions of \cite{us} for studying the designs which were found to be superior to the ones theoretically investigated in \cite{us}. We also extend our constructions for new construction schemes which provide even better coverings (in the class of nested designs) than the ones numerically found in \cite{us}. In view of a close connection of the problem of quantization to the problem of covering, we extend our theoretical approximations and practical recommendations to the problem of construction of efficient quantization designs in a cube $[-1,1]^d$. In the last section, we discuss the problems of covering and quantization in a $d$-dimensional simplex; practical significance of this problem has been communicated to the authors by Professor Michael Vrahatis, a co-editor of the present volume.
