Gradient-Enhanced Point-Collocation Polynomial Chaos Method

According to Wiener [22], f (|) can be approximated by a truncated polynomial chaos expansion (PCE)

f(l) = І Wl) (1)


where Щ is Hermite polynomial chaos (PC) to which a detailed description can be found in, e. g. [18]. The total number of terms is K = (p + d)l/(pldl) with p the order of PC.

To determine the coefficients ci we use a point-collocation method similar to the one used in [15], the difference being that we utilize gradient information. In this gradient-enhanced point-collocation polynomial chaos (GEPC) method the c = {c0, Ci, •••, cK}T is determined by solving the following system,

denote the sample points. The K is chosen to be half of the number of available “con­ditions”, N(1 + d), for the best performance according to [15]. This over-determined system is solved by a Least Squares method.

For this UQ job we first establish a GEPC surrogate model of f (|) based on QMC samples of the CFD model, and compute the mean and the variance of f (|) directly from the coefficients,

Д = co, o2 = Z(ct)2 ■ E[V2(S)] (2)


The exceedance probabilities and pdf are integrated by a large number (105) of QMC samples on the surrogate model.