Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 272

Once the coefficients have been evaluated ( Equation 6.32 ), the approximation of the ran-
dom response is the random variable

= ∑

·

PC ()

·

Y = M X y Ψ ( . X

(6.33)

α

α

α∈ A

The above equation may also be interpreted as a response surface, that is, a function

x M

PC ()

x = α∈ · α

Ψ

() x that allows one to surrogate (fast, although approximately) the

A

original model y = M ( x ).

6.3.5 Validation
6.3.5.1 Error estimators
As mentioned already, it is not possible to know in advance how to choose the maximal
polynomial degree in the standard truncation scheme ( Equation 6.18 ) . A crude approach
would consist of testing several truncation schemes of increasing degree (e.g., p = 2, 3, 4)
and observe if there is some convergence for the quantities of interest. Recently, a posteriori
error estimates have been proposed by Blatman and Sudret (2010) that allow for an objec-
tive evaluation of the accuracy of any truncated PCE.
First of all, it is reminded that a good measure of the error committed by using a truncated
series expansion is the mean-square error of the residual (which is also called generalization
error in statistical learning theory):

2

∑

def E

2

ˆ

Err

G =

ε = E Y − y Ψ α () . X (6.34)

α

α∈ A

In practice, the latter is not known analytically; yet, it may be estimated by an MCS using
a large sample set, say X val =… { x , , x }:

1 n

val

2

n

1

val

∑ = ∑

· =

def

· α Ψ α

(6.35)

Err M () x − y (). x

G

i i

n

val

i 1 α∈ A

The so-called validation set X val shall be large enough to get an accurate estimation, for
example, n val = 10 3−5 . However, as the computation of Err · requires evaluating M for each
point in X val , this is not affordable in real applications and would ruin the efficiency of the
approach. Indeed, the purpose of using PCEs is to avoid MCS, that is, to limit the number of
runs of the computational model M in Equation 6.30 to the smallest possible number, typically
n = 50 to a few hundreds.
As a consequence, to get an estimation of the generalization error ( Equation 6.34 ) at
an affordable computational cost, the points in the ED X could be used in Equation 6.35
instead of the validation set, leading to the so-called empirical error Err · defined by

2

n

def 1

∑ = ∑

· =

· α (6.36)

Err M () x

() i

− y Ψ (), x

() i

x

() i

∈ X E

n

α

i 1 α∈ A