MULTI-ELEMENT GENERALIZED POLYNOMIAL CHAOS FOR ...

912 XIAOLIANG WAN AND GEORGE EM KARNIADAKIS
π 1
π 3
π 1
π 3
4
2
0
−2
x 10−4
6
π 2
0 0.5 1
x 10 −3
−4
−6
0 0.5 1
x
x 10 −3
p=1
−0.5
p=2
−1
x
x 10−11
6
x 10−14
1.5
4
2
0
−2
π 4
1.5
1
0.5
x 10−7
2
0 0.5 1
x 10 −3
−4
−6
0 0.5 1
x
x 10 −3
−0.5
p=3 p=4
−1
x
Fig. 3.1. Orthogonal polynomials for random variable X, π1 to π4.
1
0.5
0
−0.5
π 2
p=1 p=2
−1
−1 −0.5 0 0.5 1
−0.5
−1 −0.5 0 0.5 1
y
y
0.4
0.2
0
−0.2
−0.4
p=3
−0.6
−1 −0.5 0
y
0.5 1
π 4
0
1
0.5
0
1
0.5
0
0.2
0.1
0
−0.1
p=4
−1 −0.5 0
y
0.5 1
Fig. 3.2. Orthogonal polynomials for random variable Y , π1 to π4.
3.1.2. Orthogonal polynomials for Beta distribution. From the ME-gPC
scheme, we know that the orthogonal basis in each random element depends on a
particular part of the PDF of the random inputs. We now demonstrate such a dependence
using a Beta-type random variable X with α = 1 and β = 4. For simplicity, we
consider only two random elements: [−1, 0] and [0, 1]. We define a random variable