Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. Optimization and uncertainty quantification problems typically require a large number of runs of the computational model at hand, which may not be feasible with high-fidelity models directly. Thus surrogate models (a.k.a metamodels) have been increasingly investigated in the last decade. Polynomial Chaos Expansions (PCE) and Kriging are two popular non-intrusive metamodelling techniques. PCE surrogates the computational model with a series of orthonormal polynomials in the input variables where polynomials are chosen in coherency with the probability distributions of those input variables. A least-square minimization technique may be used to determine the coefficients of the PCE. On the other hand, Kriging assumes that the computer model behaves as a realization of a Gaussian random process whose parameters are estimated from the available computer runs, i.e. input vectors and response values. These two techniques have been developed more or less in parallel so far with little interaction between the researchers in the two fields. In this paper, PC-Kriging is derived as a new non-intrusive meta-modeling approach combining PCE and Kriging. A sparse set of orthonormal polynomials (PCE) approximates the global behavior of the computational model whereas Kriging manages the local variability of the model output. An adaptive algorithm similar to the least angle regression algorithm determines the optimal sparse set of polynomials. PC-Kriging is validated on various benchmark analytical functions which are easy to sample for reference results. From the numerical investigations it is concluded that PC-Kriging performs better than or at least as good as the two distinct meta-modeling techniques. A larger gain in accuracy is obtained when the experimental design has a limited size, which is an asset when dealing with demanding computational models.

Structural reliability analysis aims at computing the probability of failure of systems whose performance may be assessed by using complex computational models (e.g. expensive-to-run finite element models). A direct use of Monte Carlo simulation is not feasible in practice, unless a surrogate model (such as Kriging, a.k.a Gaussian process modeling) is used. Such meta-models are often used in conjunction with adaptive experimental designs (i.e. design enrichment strategies), which allows one to iteratively increase the accuracy of the surrogate for the estimation of the failure probability while keeping low the overall number of runs of the costly original model. In this paper we develop a new structural reliability method based on the recently developed Polynomial-Chaos Kriging (PC-Kriging) approach coupled with an active learning algorithm known as AK-MCS. We formulate the problem in such a way that the computation of both small probabilities of failure and extreme quantiles is unified. We discuss different convergence criteria for both types of analyses, and show in particular that the original AK-MCS stopping criterion may be over-conservative. We finally elaborate a multi-point enrichment algorithm which allows us to add several points in each iteration, thus fully exploiting high-performance computing architectures. The proposed method is illustrated on three examples, namely a two-dimensional case which allows us to underline the advantages of our approach compared to standard AK-MCS. Then the quantiles of the 8-dimensional borehole function are estimated. Finally the reliability of a truss structure (10 random variables) is addressed. In all case, accurate results are obtained with about 100 runs of the original model.

In modern engineering, computer simulations are a popular tool to analyse, design, and optimize systems. Furthermore, concepts of uncertainty and the related reliability analysis and robust design are of increasing importance. Hence, an efficient quantification of uncertainty is an important aspect of the engineer's workflow. In this context, the characterization of uncertainty in the input variables is crucial. In this paper, input variables are modelled by probability-boxes, which account for both aleatory and epistemic uncertainty. Two types of probability-boxes are distinguished: free and parametric (also called distributional) p-boxes. The use of probability-boxes generally increases the complexity of structural reliability analyses compared to traditional probabilistic input models. In this paper, the complexity is handled by two-level approaches which use Kriging meta-models with adaptive experimental designs at different levels of the structural reliability analysis. For both types of probability-boxes, the extensive use of meta-models allows for an efficient estimation of the failure probability at a limited number of runs of the performance function. The capabilities of the proposed approaches are illustrated through a benchmark analytical function and two realistic engineering problems.

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Global sensitivity analysis aims at determining which uncertain input parameters of a computational model primarily drives the variance of the output quantities of interest. Sobol' indices are now routinely applied in this context when the input parameters are modelled by classical probability theory using random variables. In many practical applications however, input parameters are affected by both aleatory and epistemic (so-called polymorphic) uncertainty, for which imprecise probability representations have become popular in the last decade. In this paper, we consider that the uncertain input parameters are modelled by parametric probability boxes (p-boxes). We propose interval-valued (so-called imprecise) Sobol' indices as an extension of their classical definition. An original algorithm based on the concepts of augmented space, isoprobabilistic transforms and sparse polynomial chaos expansions is devised to allow for the computation of these imprecise Sobol' indices at extremely low cost.In particular, phantoms points are introduced to build an experimental design in the augmented space (necessary for the calibration of the sparse PCE) which leads to a smart reuse of runs of the original computational model. The approach is illustrated on three analytical and engineering examples which allows one to validate the proposed algorithms against brute-force double-loop Monte Carlo simulation. Keywords: uncertainty quantification --global sensitivity analysis --probability-boxes --imprecise Sobol' indices -sparse polynomial chaos expansions 1 Introduction Computational simulation tools, such as finite element models (FEM), are the most popular approach to model complex systems and processes in modern engineering. Such simulation tools map a set of input parameters describing the system and its environmental and operational conditions through a computational model to a so-called quantity of interest (QoI), e.g. performance indicators. The input parameters are often not perfectly known, due to noisy measurements, expert judgement, or intrinsic variability. Hence, each variable is modelled by an uncertainty model, as an example by a probability distribution. The uncertainty propagates through the computational model and results in an uncertain QoI. Sensitivity analysis (SA) examines the impact of the uncertainty in the input vector of a computational model onto the uncertainty in the QoI. This is of importance in practice where the relation 1 arXiv:1705.10061v1 [stat.CO]