Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 277

(a) (b) (c)

Figure 6.4 Sketch of the different truncation sets (black full circles). (a) Standard truncation A M,p , (b) hyper-
bolic truncation A M,p,q , (c) sparse truncation A .

best one-term expansion. Then the current approximation is improved by moving along the
direction ( + up to a point where the residual becomes equicorrelated with a third

polynomial Ψ α 3 , and so on.
In the end, the LAR algorithm has produced a sequence of less and less sparse expansions.
The LOO error of each expansion can be evaluated by Equation 6.43 . The sparse model
providing the smallest error is retained. The great advantage of LAR is that it can also be
applied in the case when the size of the candidate basis A is larger than the cardinality of
the experimental design, card X. Usually, the size of the optimal sparse truncation is smaller
than card X in the end. Thus, the coefficients of the associated PC expansion may be recom-
puted by least-square minimization for a better accuracy (Efron et al., 2004).
Note that all the above calculations are conducted from a prescribed initial ED X. It may
be that this size is too small to address the complexity of the problem, meaning that there is
not enough information to find a sparse expansion with a sufficiently small LOO error. In
this case, overfitting appears, which can be detected automatically as shown in Blatman and
Sudret (2011a). At that point, the ED shall be enriched by adding new points (Monte Carlo
samples or nested LHS).
All in all, a fully automatic “basis-and-ED” adaptive algorithm may be devised that solely
requires to prescribe the target accuracy of the analysis, that is, the maximal tolerated LOO
error, and an initial ED. The algorithm then automatically runs LAR analysis with increas-
ingly larger candidate sets A, and possibly by increasing large EDs so as to reach the pre-
scribed accuracy (see Figure 6.5 ) . Note that extensions to vector-valued models have been
recently proposed in Blatman and Sudret (2011b, 2013).

ΨΨ
α

)

α

1 2

6.4 PoSt-ProCeSSIng For engIneerIng aPPlICatIonS

The PCE technique presented in the previous sections leads to cast the QoI of a computa-
tional model Y = M( X ) through a somewhat abstract representation by a polynomial series
ˆ

α∈A αα X ( Equation 6.33 ) . Once the PC basis has been set up ( a priori or using an
adaptive algorithm such as LAR) and once the coefficients have been calculated, the series
expansion shall be post-processed so as to provide engineering-wise meaningful numbers
and statements: what is the mean behavior of the system (mean QoI), scattering (variance
of the QoI), confidence intervals, or probability of failure (i.e., the probability that the QoI
exceeds an admissible threshold)? In this section, the various ways of post-processing a PC
expansion are reviewed.

ˆ

Y =Σ y Ψ ()