Dr
Dr1
k r (14)
m0 m1

m1
m2
mr 1 mr
mr
mr 1

Dr

det
mr 1 mr
m2r 2 m2r 1
(15)
mr mr 1
m2r 1 m2r
College of Mechanical and Electrical Engineering
2.2 多项式混沌展开
Hunan University of Science and Technology
m0 m1
Pr
(x)

1 Dr 1
det

m1

mr 1
m2 mr
1 x
mr 1 mr
mr
mr 1


m2r 2
m2r 1
xr1 xr

sin(x)
(1)k
x2k 1
k 0 (2k 1)!
图1：基于泰勒展开式对sin(x)的逼近
College of Mechanical and Electrical Engineering

sin(x) sin( x) k0
(1)k x2k 1 (2(k1)k1)!x2k1
0
Pk (x)，Pr (x)


Dr
Dr1
kr k r
Pk (x)，Pr (x)
0
Pr (x)，Pk (x)


Dr
Dr1
rk rk
College of Mechanical and Electrical Engineering
2.2 多项式混沌展开
Hunan University of Science and Technology
2.2 多项式混沌展开
Hunan University of Science and Technology
m0
xi , Pr (x)

1
det
m1
Dr 1
mr 1
E[xi ]
m1 m2
mr E[xi1 ]
mr 1 mr
mr
mr 1


m2r 2
m2r 1

E[y]

E
R r0
ar xr


R r 0
ar E[xr
]
(6)
mr E[xr ] xr (x)dx
(7)
R
E[ y] armr
(8)
r 0
College of Mechanical and Electrical Engineering
2.1 数值积分
Hunan University of Science and Technology

E[y]
n
ws H ts

n
R
ws artsr
R
ar


n
wstsr

(9)
s 1
s 1
r 0
r 0
s1

n
wstsr mr， r 0,1, , R
Company
LOGO
高斯型多维积分公式
报告人：肖 青 导 师：周少武
Hunan University of Science and Technology
1、研 究 背 景
内容
2、单 变 量 函 数 2.1 数 值 积 分 研 2.2 多 项 式 混 沌 展 开研 3、多 变 量 函 数 3.1 张 量 积研 3.2 稀 疏 网 格 法 3.3 容 积 量 法 3.4 算 例 4、总 结
2.Байду номын сангаас 多项式混沌展开
Hunan University of Science and Technology

cr

Dr 1 Dr
H (x)，Pr (x)
Dr1 Dr
H (x) Pr (x) (x)dx(17)
c0

1 D0
H (x) P0 (x) (x)dx
1. 研究背景
Hunan University of Science and Technology

n
E[yr ]
ws H r ts,1 , ,ts,m
(3)
s 1
College of Mechanical and Electrical Engineering
2. 单变量函数
Hunan University of Science and Technology

R 2n 1 r 0,1, , 2n 1
College of Mechanical and Electrical Engineering
2.1 数值积分
Hunan University of Science and Technology

m0 m1
det

m1
m2

mn mn1
2.1 数值积分
Hunan University of Science and Technology

(x)
1
x2
e2
2
P0 (x) 1 P1(x) x P2 (x) x2 1 P3 (x) x3 3x
Pn1(x) xPn (x) nPn1(x)
College of Mechanical and Electrical Engineering
mn
mn
1


0

m2n

College of Mechanical and Electrical Engineering
2.1 数值积分
Hunan University of Science and Technology
m0 m1
Pn (x)

1 Dn1
det

m1

m1
E det
m2
Dr 1


mr
1
mr

xi
x i 1
mr 1 mr
mr mr 1


m2r 2
m2r 1

xir 1 xir
College of Mechanical and Electrical Engineering
2.1 数值积分
Hunan University of Science and Technology

Pn (x) ( An x Bn )Pn1(x) CnPn2 (x)
(12)
College of Mechanical and Electrical Engineering
2.1 数值积分
2.2 多项式混沌展开
Hunan University of Science and Technology

xi , Pr (x) 0
xi , Pr (x)
Dr Dr 1
k
Pk (x) bi xi i0
k
Pk (x)，Pr (x) = bi i0
xi，Pr (x)

0 b0
mn1
m2 mn
1 x
mn1 mn
mn
mn1


m2 n 2
m2 n 1

xn1 xn
(11)
m0 m1
Dn1

det

m1
m2

mn
1
mn
mn1
mn


m2
n2

College of Mechanical and Electrical Engineering


y = H x cr Pr (x) r 0
(16)

H (x)，Pr (x) ck Pk (x)，Pr (x) k 0

ck Pk (x)，Pr (x) k 0

cr

Dr Dr 1
College of Mechanical and Electrical Engineering
College of Mechanical and Electrical Engineering
1. 研究背景
Hunan University of Science and Technology

输 入
非线性传 递系统
输 出
X
y
y = H x1 , ,xi , ,xm
(1)
College of Mechanical and Electrical Engineering


t n-1 n


wn

mn1
(13)
College of Mechanical and Electrical Engineering
2.1 数值积分
Hunan University of Science and Technology

n
E[ y]= H x xdx wsH ts s 1
s 1
(10)
College of Mechanical and Electrical Engineering

sin(x) sin( x) k0
(1)k x2k 1 (2(k1)k1)!x2k1