Propagation of Uncertainties Modelled by Parametric P-boxes Using Sparse Polynomial Chaos Expansions Roland Schöbi, Bruno Sudret To cite this version: Roland Schöbi, Bruno Sudret. Propagation of Uncertainties Modelled by Parametric P-boxes Using Sparse Polynomial Chaos Expansions. 12th Int. Conf. on Applications of Statistics and Probability in Civil Engineering (ICASP12), Jul 2015, Vancouver, Canada. hal-01247151 HAL Id: hal-01247151 https://hal.archives-ouvertes.fr/hal-01247151 Submitted on 21 Dec 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 Propagation of Uncertainties Modelled by Parametric P-boxes Using Sparse Polynomial Chaos Expansions Roland Schöbi Ph.D. student, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Zurich, Switzerland Bruno Sudret Professor, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Zurich, Switzerland ABSTRACT: Advanced simulations, such as finite element methods, are routinely used to model the be- haviour of physical systems and processes. At the same time, awareness is growing on concepts of struc- tural reliability and robust design. This makes efficient quantification and propagation of uncertainties in computation models a key challenge. For this purpose, surrogate models, and especially Polynomial Chaos Expansions (PCE), have been used intensively in the last decade. In this paper we combine PCE and probability-boxes (p-boxes), which describe a mix of aleatory and epistemic uncertainty. In particu- lar, parametric p-boxes allow for separation of the latter uncertainties in the input space. The introduction of an augmented input space in PCE leads to a new uncertainty propagation algorithm for p-boxes. The proposed algorithm is illustrated with two applications: a benchmark analytical function and a realistic truss structure. The results show that the proposed algorithm is capable of predicting the p-box of the response quantity extremely efficiently compared to double-loop Monte Carlo simulation. 1. INTRODUCTION uncertainty), including probability-boxes (Ferson In modern engineering sciences, computational and Ginzburg, 1996), Bayesian hierarchical mod- simulations, such as finite element modelling, have els (Gelman, 2006) and Dempter-Shafer’s evidence become wide spread. The goal is to predict the re- theory (Dempster, 1967; Shafer, 1976). These sponse of a system with respect to a set of param- frameworks are generally referred to as imprecise eters, e.g. the deflection of a beam under variable probabilities. loads. The parameters (e.g. geometries, mechanical After the input uncertainty is characterized, properties, loads) are mapped to the quantity of in- it must be propagated through a computational terest through a computational model, e.g. through model. The latter, however, is often an expensive- the governing equations of the process. to-evaluate function, which can be replaced by an It is only in recent times that the traditionally approximate model, i.e. a meta-model, to reduce deterministic model parameters have been gradu- the computational effort needed. Well-known meta- ally substituted with probability distributions that modelling techniques include Polynomial Chaos account for their uncertainty. In practice though, Expansions (Ghanem and Spanos, 2003), Gaussian data available for calibrating such distributions are process modelling (a.k.a. Kriging) (Santner et al., often too sparse, thus resulting in an extra layer of 2003) and support vector machines (Gunn, 1998). uncertainty in their parameters. Different frame- This paper describes one formulation of impre- works have been proposed to quantify the latter cise probabilities in Section2 followed by an in- lack of knowledge (epistemic uncertainty) as well troduction to Polynomial Chaos expansions in Sec- as the natural variability of the process (aleatory tion3. Finally these two ingredients are combined 1 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 in Section4 and two applications are discussed in cannot be given a precise formulation. Thus the Section5. probability-box framework accounts for aleatory as well as for epistemic uncertainty in the description 2. INPUT UNCERTAINTY of a variable X. 2.1. Probability theory The lower and upper boundaries of the CDF are Traditionally, uncertainty in engineering has been denoted by [FX ;FX ]. The true, but unknown, CDF treated with probability theory. of X lies within the boundaries for any value of Consider a probability space (W;F ;P), where W x X, i.e. F (x) FX (x) FX (x); x X. The 2 X ≤ ≤ 8 2 denotes the event space equipped with s-algebra boundaries [FX ;FX ] mark the extreme cases of FX F and probability measure P. Random variables and are thus also CDFs by definition. are denoted by capital letters X(w) : W DX R The two boundaries form an intermediate space ! ⊂ where w W. A realization of variable X is de- in the variable-CDF-graph which resembles a box 2 noted by the corresponding lower case letters, e.g. (see Figure1), hence the name probability-box. x. Several random variables compose a random T 1 vector X = [X1;:::;XM] and the corresponding re- T alizations x = [x1;:::;xM] . P−box In this context a random variable X is described 0.8 cumulative distribution function F by its (CDF) X 0.6 Plausibility which expresses the probability that X < x, i.e. X F FX (x) = P(X < x). The CDF has the properties that 0.4 Belief it is monotone non-decreasing, that it tends to zero for low values of x (FX (x ¥) = 0) and that it 0.2 ! − tends to one for high values of x (FX (x ¥) = 1). ! 0 Note that such properties are valid for continuous −3 −2 −1 0 1 2 3 as well as discrete random variables. x For continuous random variables, the derivative Figure 1: P-box – CDF of a Gaussian variable with of a CDF is the probability density function fX (x) = interval-valued mX and sX dFX (x)=dx. The PDF describes the likelihood that X is in the neighbourhood of x. Due to the fact The p-box can be interpreted in the framework that the CDF is non-decreasing, the PDF has non- of Dempster-Shafer’s evidence theory (Dempster, negative values for all x X. 1967; Shafer, 1976). The lower boundary F de- 2 X As seen in the definitions above, probability the- scribes the minimum amount of probability that ory offers a single measure (i.e. the probability must be assigned to FX (x), which corresponds to the measure) to describe variability in variable X. In belief function Bel(FX (x)) in the vocabulary of ev- other words, we assume that the variability in X is idence theory. Analogously, the upper boundary of known and quantifiable by the CDF FX and the cor- the p-box is associated with the maximum amount responding PDF fX . This describes the case where of probability that might be assigned to FX (x), or variability is treated as the only source of uncer- the plausibility function Pl(FX (x)). tainty. Note that if FX (x) = FX (x) = FX (x); x X, then the p-box is called thin and conventional8 prob-2 2.2. Probability-box ability theory can be applied. A more general formulation is given by the frame- work of probability-boxes (p-boxes) which defines 2.3. Parametric p-boxes the CDF of a variable X by its lower and upper In the literature two types of p-boxes are identified, bound distributions (Ferson and Ginzburg, 1996; namely the free p-box and the parametric p-box. In Ferson and Hajagos, 2004). The idea is that due a this paper, we focus on parametric p-boxes (also lack of knowledge (epistemic uncertainty), the CDF called distributional p-boxes). 2 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 T A parametric p-box requires knowledge about The components of X = [X1;:::;XM] are assumed the shape of the true CDF but allows for uncertainty independent for the sake of simplicity throughout in its parameters. The p-box is represented by a this paper. The model response is a random vari- family of distribution functions whose parameters able Y obtained by propagating the input random q lie within an interval. For a single variable X: vector X through the computational model M . Several techniques are available for surrogating F (x) = F (x;q ); (1) X X the expensive-to-evaluate computational model M . In the following section, Polynomial Chaos Expan- where qi [q i;q i]; i = 1;:::;nq . This construction resembles2 a Bayesian hierarchical model (Gelman, sions (Ghanem and Spanos, 2003; Sudret, 2007) 2006) in which the distribution of the parameters will be briefly introduced. q is replaced by an interval. This framework al- 3.2. Polynomial Chaos Expansion lows for a clear separation of aleatory and epistemic A well-known non-intrusive meta-modelling uncertainty: aleatory uncertainty is represented by method is Polynomial Chaos Expansion (PCE) the distribution function family and epistemic un- which approximates the computational model certainty is represented by the interval on parame- M with a finite series of polynomials orthogonal with ters q . respect to the distribution of the input variables: Figure2 illustrates a parametric p-box generated by a Gaussian random variable with mean value (PCE) Y M (X ) = aa ya (X ); (2) and standard deviation varying within the intervals ≈ ∑ a A M;p 2 mX = [ 0:5;0:5] and sX = [0:7;1:0].