MULTI-ELEMENT GENERALIZED POLYNOMIAL CHAOS FOR ...

920 XIAOLIANG WAN AND GEORGE EM KARNIADAKIS
Table 3.4
Maximum normalized errors of the variance of y1, y2, and y3 at t =20for case (i) of the
K-O problem. α =1/2 and ξ1 is of Beta distribution with α =1and β =4. (The results given by
ME-gPC with θ1 =10 −7 and polynomial order p =7are used as exact solutions.)
θ1 =10 −2 θ1 =10 −3 θ1 =10 −4 θ1 =10 −5
N Error N Error N Error N Error
p =3 24 2.98e-2 77 1.85e-3 236 3.61e-5 704 1.18e-6
p =4 21 1.54e-2 45 1.05e-3 93 3.21e-4 209 4.62e-6
p =5 16 6.25e-2 34 3.30e-3 57 1.50e-4 104 2.56e-6
Table 3.5
Maximum normalized errors of the variance of y1, y2, and y3 at t =20for case (i) of the K-O
problem. α =1/2 and ξ1 is of normal distribution. (The results given by ME-gPC with θ1 =10 −7
and polynomial order p =7are used as exact solutions.)
Length of Elements
0.6
0.5
0.4
0.3
0.2
0.1
θ1 =10 −2 θ1 =10 −3 θ1 =10 −4 θ1 =10 −5
N Error N Error N Error N Error
p =3 38 2.79e-2 98 2.21e-3 292 6.58e-5 878 1.40e-6
p =4 32 4.02e-2 62 9.07e-3 128 4.34e-4 282 1.46e-6
p =5 28 4.63e-2 42 3.28e-3 78 6.46e-5 138 2.61e-6
Beta Distribution with α=1 and β=4
0
−1 −0.8 −0.6 −0.4 −0.2 0
ξ
1
0.2 0.4 0.6 0.8 1 0
1.4
1.2
1
0.8
0.6
0.4
0.2
Length of Elements
3.5
3
2.5
2
1.5
1
0.5
Gaussian Distribution
0
−6 −5 −4 −3 −2 −1 0
ξ
1
1 2 3 4 5 6 0
Fig. 3.9. Adaptive mesh and corresponding random distribution for the K-O problem with onedimensional
random inputs. p =5and θ1 =10 −3 . Left: Beta distribution with α =1and β =4.
Right: Gaussian distribution.
Error
10 −1
10 −2
10 −3
10 −4
10 −5
10 3
10 −6
MC: n −0.5
ME−gPC: p=3
ME−gPC: p=4
ME−gPC: p=5
log(n)
10 4
Error
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
MC: n −0.5
ME−gPC: p=3
ME−gPC: p=4
ME−gPC: p=5
10 4
log(n)
Fig. 3.10. Speedup for the K-O problem with one-dimensional random inputs at t =20. Left:
Beta distribution with α =1and β =4. Right: Gaussian distribution.
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05