Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 274

Blatman (2009, Appendix D)) as follows. The predicted residual in Equation 6.38 even-
tually reads:

MM MM

() x

() i

−
−

PC

( x

() i

() i PC \ i ( ) i

∆ i = () x − () x = , (6.42)

1

h

where h i is the i ith diagonal term of matrix a(a T a) −1 a T . The LOO error estimate eventually
reads

2

n

1 MM () x

() i

−
−

PC

() , x

() i

∑

·

Err = (6.43)

LOO

n 1 h

i

i = 1

where M PC has been built up from the full ED. As a conclusion, from a single resolution
of a least-square problem using the ED X, a fair error estimate of the mean-square error is
available a posteriori using Equation 6.43 . Note that in practice, a normalized version of
the LOO error is obtained by dividing Err LO · by the sample variance Var [ Y ]. A correction
factor that accounts for the limit size of the ED is also added (Chapelle et al., 2002) that
eventually leads to

2

n

1 + tr C

Var

1 − 1

1 MM () x

() i

−
−

PC

() x

() i

∑

n emp

ˆ

LOO = , (6.44)

[]

nP − 1 h

Y i

i = 1

T

where tr(⋅) is the trace, P = card A , and C = () 1 ΨΨ / n .

emp

6.3.6 Curse of dimensionality
Common engineering problems are solved with computational models having typically
M = 10-50 input parameters. Even when using a low-order PC expansion, this leads to a
large number of unknown coefficients (size of the truncation set A M,p ), which is equal to, for
example, 286 − 23,426 terms when choosing p = 3. As explained already, the suitable size
of the ED shall be 2-3 times those numbers, which may reveal as unaffordable when the
computational model M is, for example, a finite-element model.
On the other hand, most of the terms in this large truncation set A M,p correspond to
polynomials representing interactions between input variables. Yet, it has been observed in
many practical applications that only the low-interaction terms have coefficients that are
significantly nonzero. Taking again the example { M = 50, p = 3}, the number of univari-
ate polynomials in the input variables (i.e., depending only on U 1 , on U 2 , etc.) is equal to
p ⋅ M = 150, that is, <1% of the total number of terms in A 50,3 . As a conclusion, the com-
mon truncation scheme in Equation 6.18 leads to compute a large number of coefficients,
whereas most of them may reveal negligible once the computation has been carried out.
Owing to this sparsity-of-effect principle, Blatman (2009) has proposed to use truncation
schemes that favor the low-interaction (also called low-rank) polynomials. Let us define the
rank r of a multi-index α by the number of nonzero integers in α, that is,

M

∑

def α

r = = 1 {

} .

(6.45)

α > 0

0

i

i = 1