The application of polynomial chaos expansions (PCEs) to the propagation of uncertainties in stochastic dynamical models is well-known to face challenging issues. The accuracy of PCEs degenerates quickly in time. Thus maintaining a sufficient level of long term accuracy requires the use of high-order polynomials. In numerous cases, it is even infeasible to obtain accurate metamodels with regular PCEs due to the fact that PCEs cannot represent the dynamics. To overcome the problem, an original numerical approach was recently proposed that combines PCEs and non-linear autoregressive with exogenous input (NARX) models, which are a universal tool in the field of system identification. The approach relies on using NARX models to mimic the dynamical behaviour of the system and dealing with the uncertainties using PCEs.The PC-NARX model was built by means of heuristic genetic algorithms. This paper aims at introducing the least angle regression (LAR) technique for computing PC-NARX models, which consists in solving two linear regression problems. The proposed approach is validated with structural mechanics case studies, in which uncertainties arising from both structures and excitations are taken into account. Comparison with Monte Carlo simulation and regular PCEs is also carried out to demonstrate the effectiveness of the proposed approach.Keywords: surrogate models -polynomial chaos expansions -nonlinear autoregressive with exogenous input (NARX) models -Monte Carlo simulation -dynamical systems 1 arXiv:1604.07627v1 [stat.ME] 26 Apr 2016Modern engineering and applied sciences have greatly benefited from the rapid increase of available computational power. Indeed, computational models allow for an accurate representation of complex physical phenomena, e.g. fluid and structural dynamics. Physical systems are in nature prone to aleatory uncertainties, such as the natural variability of system properties, excitations and boundary conditions. In order to obtain reliable predictions from numerical simulations, it is of utmost importance to take into account these uncertainties. In this context, uncertainty quantification has gained particular interest in the last decade. The propagation of uncertainties from the input to the output of the model is commonly associated with sampling-based methods, e.g. Monte Carlo simulation, which are robust, however not suitable when only a small number of simulations is affordable or available. Metamodelling techniques can be employed to circumvent this issue. Polynomial chaos expansions (PCEs) (Ghanem and Spanos, 2003;Soize and Ghanem, 2004) are a powerful metamodelling technique that is widely used in numerous disciplines. However, it is well-known that PCEs exhibit difficulties when being applied to stochastic dynamical models (Wan and Karniadakis, 2006a;Gerritsma et al., 2010).Many efforts have been focused on solving this problem. Wan and Karniadakis (2006a,b) proposed multi-element PCEs, which rely on the decomposition of the random input space into sub-domains and the local comp...

a b s t r a c tIn the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol' indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The derivative-based global sensitivity measures (DGSMs) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. Closed-form expressions for Hermite, Legendre and Laguerre polynomial expansions are given. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis.

Polynomial chaos expansions (PCE) have proven efficiency in a number of fields for propagating parametric uncertainties through computational models of complex systems, namely structural and fluid mechanics, chemical reactions and electromagnetism, etc. For problems involving oscillatory, time-dependent output quantities of interest, it is well-known that reasonable accuracy of PCE-based approaches is difficult to reach in the long term. In this paper, we propose a fully non-intrusive approach based on stochastic time warping to address this issue: each realization (trajectory) of the model response is first rescaled to its own time scale so as to put all sampled trajectories in phase in a common virtual time line. Principal component analysis is introduced to compress the information contained in these transformed trajectories and sparse PCE representations using least angle regression are finally used to approximate the components.The approach shows remarkably small prediction error for particular trajectories as well as for second-order statistics of the latter. It is illustrated on different benchmark problems well known in the literature on time-dependent PCE problems, ranging from rigid body dynamics, chemical reactions to forced oscillations of a non linear system.Keywords: surrogate models -sparse polynomial chaos expansions -stochastic ordinary differential equations -stochastic time warping -dynamical systems 1 arXiv:1609.09286v2 [stat.CO]

Keywords: hierarchical adaptive polynomial chaos expansions, heredity principle, sparsity principle, hierarchy principle, high dimensional problems.Abstract. Polynomial chaos expansions (PCE) are widely used in the framework of uncertainty quantification. However, when dealing with high dimensional complex problems, challenging issues need to be faced. For instance, high-order polynomials may be required, which leads to a large polynomial basis whereas usually only a few of the basis functions are in fact significant. Taking into account the sparse structure of the model, advanced techniques such as sparse PCE (SPCE), have been recently proposed to alleviate the computational issue. In this paper, we propose a novel approach to SPCE, which allows one to exploit the model's hierarchical structure. The proposed approach is based on the adaptive enrichment of the polynomial basis using the so-called principle of heredity. As a result, one can reduce the computational burden related to a large pre-defined candidate set while obtaining higher accuracy with the same computational budget.

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