Surrogate models for uncertainty quantification in the context of imprecise probability modelling

Abstract

Nowadays, computer simulations are a popular engineering tool to design systems of even increa- sing complexity and to determine their performance. Theses simulations aim at reproducing the physical process at hand and hence provide a solution to the underlying governing equations. As an example, finite element models have become a standard tool in modern civil and mechanical engineering. Such models exploit the available computer power, meaning that a single run of the model can take up to hours of days of computing.

At the same time, uncertainties are intrinsic to theses processes and simulation methods. Typically model parameters are not known deterministically, but inferred from data and then modelled probabilistically. However, a common situation in practice is to have a limited budget for data acquisition and hence to end up with sparse datasets. This introduces epistemic uncertainty (lack of knowledge) alongside aleatory uncertainty (natural variability). The mix of these two sources of uncertainties is often referred to as imprecise probabilities.

Among the various concepts to describe imprecise probabilities, there are probability-boxes (p-boxes), which model the uncertainty by an interval-valued cumulative distribution function. Hence, p-boxes are a generalization of the popular probability theory. P-boxes provide an intuitive framework that is more general than conventional probability theory and that allows for a clear distinction between aleatory and epistemic uncertainty. Due to its intuitive setup, p-boxes are easy to interpret and hence easy to apply to practical problem settings.

In this context, engineers are concerned with uncertainty quantification. In other words, engineers analyse how the uncertainty in the model parameters affects the uncertainty in the quantity of interest. In this sense, uncertainty quantification includes uncertainty propagation, structural reliability analysis, sensitivity analysis, and design optimization.

Due to the complexity of the computer simulations and the use of p-boxes, however, these analyses may become intractable. In order to reduce the computational costs and to make uncertainty quantification analyses tractable, meta-models are used throughout this thesis. Kriging (a.k.a Gaussian process modelling) and polynomial chaos expansions (PCE) are two state-of- the-art meta-modelling algorithms which approximate the complex computer simulations with an easy-to-evaluate function. Furthermore, the proposed Polynomial-Chaos-Kriging approach further increases the accuracy of the approximation when comparing to Kriging and PCE.

Depending on the uncertainty quantification analysis, the meta-modelling approaches are modified to fit the needs of the specific analysis in the context of p-boxes. For uncertainty propagation and structural reliability analysis, two-level meta-modelling approaches are proposed. The use of meta-models at different stages of the analysis allows for an efficient, i.e. accurate and inexpensive, estimation of the quantity of interest. Emphasis is taken on the limitation of computational resources, i.e. of the number of runs of the complex computer simulations.

The approaches are validated on a number of benchmark function, ranging from purely analytical function to finite element models. Further, case studies from the field of aeronautics and geotechnical engineering show the applicability of the proposed algorithm in realistic and complex engineering settings. The variety of examples shows the flexibility and versatility of the proposed algorithms. Hence, the proposed approaches are of importance for the engineering practice when the standard probabilistic approach fails to characterize the uncertainty. 

Keywords

Imprecise probabilities, Kriging, PC-Kriging, polynomial chaos expansions, probability-boxes, sensitivity analysis, structural reliability analysis.

BibTeX cite

@PHDTHESIS{SchoebiThesis,
 author = {Sch\"obi, R.},
 title = {Surrogate models for uncertainty quantification in the context of imprecise probability modelling},
 school = {ETH Z\"urich, Z\"urich, Switzerland},
 year = {2017}
}