高斯型多维积分公式PPT课件

1、1,高斯型多维积分公式,报告人：肖 青 导 师：周少武,.,2,Hunan University of Science and Technology,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,.,3,Hunan University of Science and Technology,1. 研究背景,(1),.,4,Hunan University of Science and Technolog

2、y,1. 研究背景,(2),.,5,Hunan University of Science and Technology,1. 研究背景,(3),.,6,Hunan University of Science and Technology,2. 单变量函数,(4),(5),.,7,Hunan University of Science and Technology,2. 单变量函数,(6),(7),(8),.,8,Hunan University of Science and Technology,2.1 数值积分,(9),(10),.,9,Hunan University of Scienc

3、e and Technology,2.1 数值积分,图1：基于泰勒展开式对sin(x)的逼近,.,10,Hunan University of Science and Technology,2.1 数值积分,图2：基于Hermite多项式对sin(x)的逼近,.,11,Hunan University of Science and Technology,2.1 数值积分,.,12,Hunan University of Science and Technology,2.1 数值积分,.,13,Hunan University of Science and Technology,2.1 数值积分

4、,(11),.,14,Hunan University of Science and Technology,2.1 数值积分,.,15,Hunan University of Science and Technology,2.1 数值积分,(12),.,16,Hunan University of Science and Technology,2.1 数值积分,(13),.,17,Hunan University of Science and Technology,2.1 数值积分,.,18,Hunan University of Science and Technology,2.2 多项式混

5、沌展开,(14),(15),.,19,Hunan University of Science and Technology,2.2 多项式混沌展开,.,20,Hunan University of Science and Technology,2.2 多项式混沌展开,.,21,Hunan University of Science and Technology,2.2 多项式混沌展开,.,22,Hunan University of Science and Technology,2.2 多项式混沌展开,.,23,Hunan University of Science and Technolog

6、y,2.2 多项式混沌展开,.,24,Hunan University of Science and Technology,2.2 多项式混沌展开,(16),.,25,Hunan University of Science and Technology,2.2 多项式混沌展开,(17),(18),(19),.,26,Hunan University of Science and Technology,2.2 多项式混沌展开,基于式（16）计算的数学期望，有：,(20),.,27,Hunan University of Science and Technology,2.2 多项式混沌展开,(21

7、),.,28,Hunan University of Science and Technology,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),.,29,Hunan University of Science and Technology,3. 多变量函数,(23),(24),(25),(26),.,30,Hunan University of Scienc

8、e and Technology,3.1 张量积,(27),(28),.,31,Hunan University of Science and Technology,3.1 张量积,表1：Gauss-Hermite积分的节点和权重,.,32,Hunan University of Science and Technology,3.1 张量积,.,33,Hunan University of Science and Technology,3.1 张量积,.,34,Hunan University of Science and Technology,3.2 稀疏网格法,(29),(30),.,35

9、,Hunan University of Science and Technology,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。,.,36,Hunan University of Science and Technology,3.3 容积量法,(31),.,37,Hunan University of Science and Technology,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),.,38,Hun

10、an University of Science and Technology,3.4 算例,(33),表3：不同积分法的计算结果,.,39,Hunan University of Science and Technology,3.4 算例,(34),表4：不同积分法的计算结果,.,40,Hunan University of Science and Technology,3.4 算例,(35),表5：不同积分法的计算结果,.,41,Hunan University of Science and Technology,3.4 算例,(36),表6：不同积分法的计算结果,.,42,Hunan University of Science and Technology,4. 总 结,高斯型多维积分公式，就是对单维积