论文标题
高维多型的立方体的消除节点消除算法
A Node Elimination Algorithm for Cubature of High-Dimensional Polytopes
论文作者
论文摘要
节点消除是一种获得多元积分近似值的立方体规则的数值方法。从已知的立方体规则开始，选择节点以消除，然后通过迭代求解矩方程来构建一个新的，更有效的规则。本文介绍了一个新的标准，用于选择基于矩方程的线性化来消除哪些节点。此外，还引入了受惩罚的迭代求解器，以确保权重为正，并且节点在集成域内。描述了在几个空间维度中为各种多面构建初始正交规则的策略。提出了两个，三维和四个维度的高效率规则。将新规则与通过将一维正交规则和域转换的张量产品以及已知的分析构造的立方体规则相结合而获得的规则进行了比较。
Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. This paper introduces a new criterion for selecting which nodes to eliminate that is based on a linearization of the moment equation. In addition, a penalized iterative solver is introduced, that ensures that weights are positive and nodes are inside the integration domain. A strategy for constructing an initial quadrature rule for various polytopes in several space dimensions is described. High efficiency rules are presented for two, three and four dimensional polytopes. The new rules are compared with rules that are obtained by combining tensor products of one dimensional quadrature rules and domain transformations, as well as with known analytically constructed cubature rules.
