7.5 gPC and PCM
167
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-40000
-20000
0
20000
40000
60000
0
5
10
15
20
25
30
35
Three conductivity profiles for L/delta=3
Fig. 7.5 Three sample functions for the conductivity profile when L/δ = 3. These functions are
the departure from the mean value of σ host = 3.02 × 10 5 S/m. We assume a uniform probability
density function, centered at zero and with variance = 1, for the random variables, {ξ i }, in the
Karhunen-Loève expansion
∞
−∞
· · ·
∞
−∞
Z(l, ξ 1 , · · · , ξ M )π M (ξ 1 , · · · , ξ M )dξ 1 · · · dξ M
=
∞
−∞
· · ·
∞
−∞
Z(l, ξ 1 , · · · , ξ M )π(ξ 1 ) · · · π(ξ M )dξ 1 · · · dξ M , (7.21)
where we are assuming that the {ξ i }, i = 1, · · · , M, are independent, identically
distributed random variables (iid), with the common density function, π(ξ ). This
allows us to replace the joint M−dimensional density function, π M , with the
product of M copies of the univariate density function, π , in order to make the
following developments numerically feasible. Furthermore, we are free to decide
on π to suit the purposes of our problem; the K-L expansion specifies only that the
{ξ i } are uncorrelated.
The generalized polynomial chaos (gPC) expansion of degree N for
Z(l, ξ 1 , · · · , ξ M ) is given by Xiu [143]:
Z(l, ξ 1 , · · · , ξ M ) =
0≤|i|≤N
ˆ
Z i 1 (l) · · · ˆ
Z i M (l)φ i 1 (ξ 1 ) · · · φ i M (ξ M ) ,
(7.22)
where |i| = i 1 +· · ·+i M and {φ k (ξ i )} N
k=0 is the set of univariate gPC basis functions
in ξ i of degree 0 ≤ k ≤ N. They are orthogonal polynomials associated with the
density function, π(ξ ), in the sense that
167
-80000
-60000
-40000
-20000
0
20000
40000
60000
0
5
10
15
20
25
30
35
Three conductivity profiles for L/delta=3
Fig. 7.5 Three sample functions for the conductivity profile when L/δ = 3. These functions are
the departure from the mean value of σ host = 3.02 × 10 5 S/m. We assume a uniform probability
density function, centered at zero and with variance = 1, for the random variables, {ξ i }, in the
Karhunen-Loève expansion
∞
−∞
· · ·
∞
−∞
Z(l, ξ 1 , · · · , ξ M )π M (ξ 1 , · · · , ξ M )dξ 1 · · · dξ M
=
∞
−∞
· · ·
∞
−∞
Z(l, ξ 1 , · · · , ξ M )π(ξ 1 ) · · · π(ξ M )dξ 1 · · · dξ M , (7.21)
where we are assuming that the {ξ i }, i = 1, · · · , M, are independent, identically
distributed random variables (iid), with the common density function, π(ξ ). This
allows us to replace the joint M−dimensional density function, π M , with the
product of M copies of the univariate density function, π , in order to make the
following developments numerically feasible. Furthermore, we are free to decide
on π to suit the purposes of our problem; the K-L expansion specifies only that the
{ξ i } are uncorrelated.
The generalized polynomial chaos (gPC) expansion of degree N for
Z(l, ξ 1 , · · · , ξ M ) is given by Xiu [143]:
Z(l, ξ 1 , · · · , ξ M ) =
0≤|i|≤N
ˆ
Z i 1 (l) · · · ˆ
Z i M (l)φ i 1 (ξ 1 ) · · · φ i M (ξ M ) ,
(7.22)
where |i| = i 1 +· · ·+i M and {φ k (ξ i )} N
k=0 is the set of univariate gPC basis functions
in ξ i of degree 0 ≤ k ≤ N. They are orthogonal polynomials associated with the
density function, π(ξ ), in the sense that