Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 271

that is

2

∑

y = arg min

card

E M ( X ) − y Ψ ( X ) . (6.28)

αα

A

y ∈

α∈ A

The residual in Equation 6.27 is nothing but a quadratic function of the (still unknown)
coefficients {}

y αα∈A . By simple algebra, it can be proven that the solution is identical to that
obtained by projection in Equation 6.25 . However, the setup in Equation 6.28 that is simi-
lar to regression opens to new computational schemes.
For this purpose, the discretized version of the problem is obtained by replacing the
expectation operator in Equation 6.28 by the empirical mean over a sample set:

2

n

1

∑ ∑

( ) −

ˆ

() i () i

(6.29)

y = argmin M x y

Ψ

(). x

α

n card A y ∈

i = 1 α∈ A

In this expression, χ = { x ( i ), i = 1, …, n } is a sample set of points (also called experimental
design [ED]), that is typically obtained by MCS of the input random vector X . The least-
square minimization problem in Equation 6.29 is solved as follows:

• The computational model M is run for each point in the ED, and the results are stored
in a vector

{ }

T

() 1 () 1

…

y n

() () n

(6.30)

Y M M = y = ( , x , = ( x ) .

• The information matrix is calculated from the evaluation of the basis polynomials
onto each point in the ED:

Š '

def

a = a ij = Ψ x () ) (

i

, i = 1 , … ,, nj = 1 , … , card A .

j

(6.31)

• The solution of the least-square minimization problem reads

ˆ

y = ( aa a

T1

)

− T Y

. (6.32)

To be well posed, the least-square minimization requires that the number of unknown
P = card A is smaller than the size of the ED n = card X. The empirical thumb rule
n ≈ 2 P − 3 P is often mentioned (Sudret, 2007; Blatman, 2009). To overcome the poten-
tial ill-conditioning of the information matrix, a singular-value decomposition shall be
used (Press et al., 2001).
The points used in the ED may be obtained from crude MCS. However, other types of
designs are of common use, especially Latin Hypercube sampling (LHS), see McKay et al.
(1979), or quasi-random sequences such as the Sobol' or Halton sequence (Niederreiter,
1992). From the author's experience, the latter types of design provide rather equivalent
accuracy in terms of the resulting mean-square error, for the same sample size n . Note that
deterministic designs based on the roots of the orthogonal polynomials have also been pro-
posed earlier in Berveiller et al. (2006) based on Isukapalli (1999).