高斯型多维积分公式ppt课件

高斯型多维积分公式,报告人：肖青导师：周少武,.,HunanUniversityofScienceandTechnology,内容,4、总结,3、多变量函数,1、研究背景,2、单变量函数,2.1数值积分研,3.1张量积研,3.2稀疏网格法,2.2多项式混沌展开研,3.3容积量法,3.4算例,.,HunanUniversityofScienceandTechnology,1.研究背景,(1),.,HunanUniversityofScienceandTechnology,1.研究背景,(2),.,HunanUniversityofScienceandTechnology,1.研究背景,(3),.,HunanUniversityofScienceandTechnology,2.单变量函数,(4),(5),.,HunanUniversityofScienceandTechnology,2.单变量函数,(6),(7),(8),.,HunanUniversityofScienceandTechnology,2.1数值积分,(9),(10),.,HunanUniversityofScienceandTechnology,2.1数值积分,图1：基于泰勒展开式对sin(x)的逼近,.,HunanUniversityofScienceandTechnology,2.1数值积分,图2：基于Hermite多项式对sin(x)的逼近,.,HunanUniversityofScienceandTechnology,2.1数值积分,.,HunanUniversityofScienceandTechnology,2.1数值积分,.,HunanUniversityofScienceandTechnology,2.1数值积分,(11),.,HunanUniversityofScienceandTechnology,2.1数值积分,.,HunanUniversityofScienceandTechnology,2.1数值积分,(12),.,HunanUniversityofScienceandTechnology,2.1数值积分,(13),.,HunanUniversityofScienceandTechnology,2.1数值积分,.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,(14),(15),.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,(16),.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,(17),(18),(19),.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,基于式（16）计算的数学期望，有：,(20),.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,(21),.,HunanUniversityofScienceandTechnology,2.2多项式混沌展开,若计算二阶原点矩，数值积分的算法为：对比可知：在求取二阶原点矩时，数值积分更方便。此外，延用这种思想可以方便求取更高阶的原点矩；若采用多项式混沌展开法，用系数表示输出量的高阶矩，计算很繁琐。或许说，在计算高阶矩时，数值积分对信息的处理更有效率。,(22),.,HunanUniversityofScienceandTechnology,3.多变量函数,(23),(24),(25),(26),.,HunanUniversityofScienceandTechnology,3.1张量积,(27),(28),.,HunanUniversityofScienceandTechnology,3.1张量积,表1：Gauss-Hermite积分的节点和权重,.,HunanUniversityofScienceandTechnology,3.1张量积,.,HunanUniversityofScienceandTechnology,3.1张量积,.,HunanUniversityofScienceandTechnology,3.2稀疏网格法,(29),(30),.,HunanUniversityofScienceandTechnology,3.2稀疏网格法,表2：稀疏网格法的计算量,优点：代数精度较高、计算量较低。缺点：仍不适用于较多的变量。,.,HunanUniversityofScienceandTechnology,3.3容积量法,(31),.,HunanUniversityofScienceandTechnology,3.3容积量法,为满足所有方程，有：计算量：优点：若要求的代数精度较低时，计算量较低。缺点：不适用于高阶精度，且目前尚无较好的配点方法。,(32),.,HunanUniversityofScienceandTechnology,3.4算例,(33),表3：不同积分法的计算结果,.,HunanUniversityofScienceandTechnology,3.4算例,(34),表4：不同积分法的计算结果,.,HunanUniversityofScienceandTechnology,3.4算例,(35),表5：不同积分法的计算结果,.,HunanUniversityofScienceandTechnology,3.4算例,(36),表6：不同积分法的计算结果,.,HunanUniversityofScienceandTechnology,4.总结,高斯型多维积分公式，就是对单维积分节点、权重进行排列组合，构