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.
翻译:节点清除是一种数字方法, 以获得多变量构件近似的幼稚规则。 从已知的幼稚规则开始, 选择节点来消除, 并且通过迭代解答瞬间方程式来构建新的、 效率更高的规则。 本文为选择哪些节点来消除以瞬间方程式线性化为基础的节点引入了新的标准。 此外, 引入了受罚的迭代求解器, 以确保加权为正数, 节点在整合域内。 描述了为多个空间维度的多个顶点构建初始二次二次曲线规则的战略 。 为两个、 三和四维多维的顶点提出了高效率规则 。 新规则与通过将一个维的四方形规则和域变形变形的发光产品以及已知的分析构建的二次曲线规则进行比较 。