A Kriging and Stochastic Collocation ensemble for uncertainty quantification in engineering applications

Abstract

We propose a new surrogate modeling approach by combining two non-intrusive techniques: Kriging and Stochastic Collocation. The proposed method relies on building a sufficiently accurate Stochastic Collocation model which acts as a basis to construct a Kriging model on the residuals, to combine the accuracy and efficiency of Stochastic Collocation methods in describing stochastic quantities with the flexibility and modeling power of Kriging-based approaches. We investigate and compare performance of the proposed approach with state-of-art techniques over benchmark problems and practical engineering examples on various experimental designs.

Appendix: Smolyak algorithm

The sparse interpolant $\mathbf {A}_{L,d}$ given by the Smolyak algorithm is [54]

$$\begin{aligned} \mathbf {A}_{L,d}\left( \mathbf {\varvec{\xi }}\right) = \sum _{L-d+1\le |\mathbf {k}|\le L} \left( -1\right) ^{L-|\mathbf {k}|} \left( \begin{array}{c} {d-1}\\ {L-|\mathbf {k}|} \end{array}\right) \left( U^{k_{1}} \otimes \cdots \otimes U^{k_{d}}\right) \end{aligned}$$

(17)

where $A_{L,d}$ is the weighted sum of d dimensional product rule, the vector $\mathbf {k}$ is formed by the interpolation level or order used for each variable, here $|\mathbf {k}| = k_{1} + \cdots + k_{d}$, and L is the maximum level assumed for the sparse grid. In the above expression, the desired interpolant $A_{L,d}$ is formed by combination of the one-dimensional rules $U^{k_{i}}$ of order $k_i$ which sum or total order $|\mathbf {k}|$ never exceeds the maximum level L.

Fig. 10

Two-dimensional tensor product based on 17 samples for each variable and corresponding Smolyak sparse grid $\mathbf {H}_{4,2}$ based on the Clenshaw Curtis rule

To form an interpolant $\mathbf {A}_{L,d}$ in Eq. (17), the total number of points ( $\mathbf {H}_{L,d}$) used by the interpolant is given by the following:

$$\begin{aligned} \mathbf {H}_{L,d} = \bigcup _{{L-d+1\le |\mathbf {k}|\le L}} \Theta _{1}^{k_{1}} \times \cdots \times \Theta _{d}^{k_d} \end{aligned}$$

(18)

where $\Theta$ denotes the set of points used in the one-dimensional function interpolation. Moreover, by choosing a suitable one-dimensional node scheme, e.g., Chebchev points, the set of collocation points $\Theta ^{k}$ obtained are nested.

To illustrate the grid construction based on the tensor product and sparse grid, a two-dimensional example is used here. In particular, the Clenshaw–Curtis rule is adopted to choose the node for the interpolation in each dimension: the resulting collocation point is the extrema of the Chebyshev polynomials. The total number of points $(\mathbf {H}_{4,2})$ using a level 4 sparse grid is obtained by Eq. (18). As result, a maximum of 17 nodes are chosen for each dimension and a total of 65 collocation points (Fig. 10b) are required to build the desired SC model. The corresponding tensor product grid is obtained by the product of the 17 nodes chosen in each dimension by the Clenshaw–Curtis rule. As a result, 289 $(17 \times 17)$ points (Fig. 10a) are required to build the desired SC model by a tensor product, which is approximately 4.5 times the total number of points required by the corresponding sparse grid.