ABSTRACT
The point-wise discretization error is evaluated by means of (7) and (8) (or alternatively (9)), where the variance Cˆww(x, x) and the covariance Cˆww(x1, x2) of the approximated random field are computed using the relationships (5) and (6). Two global error estimators, denoted by εM ,σ 2w and εM ,Cww , are defined as the integral of the discretization error estimators εM ,σ 2w (x) and εM ,Cww (x1, x2) over the domain D normalized to a function of the size of the length LD of the domain D:
εM ,σ 2w = 1
LD
∫
εM ,σ 2w (x)dx (10)
εM ,Cww = 1
∫
∫
εM ,Cww (x1, x2)dx1dx2 (11)
In the field of structural engineering, homogeneous random fields with constant variance σ 2w are often used in the modelling of structural properties. Under this hypothesis, the discretization error estimator (7) concerning the variance can be written as follows:
εM ,σ 2w (x) = 1 − ∑M
(12)
The analytical expression of the corresponding global error estimator (10) is:
εM ,σ 2w = 1
LD
∫
εM ,σ 2w (x)dx = 1 − ∑M
(13)
It can be shown numerically that εM ,σ 2w is much larger than εM ,Cww for a given random field and therefore the global error estimator (13) with respect to the variance should be considered as the main indicator of the accuracy of the random field discretization. As result, the eigenvalues {λi}Mi=1, the variance σ 2w and the length LD enable the quantitative assessment of the quality of the approximation of a random field w(x, θ) by means of a set of M random variables {ξi(θ)}Mi=1.
