International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Propagation of uncertainties modelled by parametric p-boxes using sparse Polynomial Chaos Expansions Schöbi, Roland; Sudret, Bruno Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Propagation of Uncertainties Modelled by Parametric P-boxes UsingSparse Polynomial Chaos ExpansionsRoland SchöbiPh.D. student, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Zurich,SwitzerlandBruno SudretProfessor, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Zurich,SwitzerlandABSTRACT: 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 uncertaintiesin computation models a key challenge. For this purpose, surrogate models, and especially PolynomialChaos Expansions (PCE), have been used intensively in the last decade. In this paper we combine PCEand 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 introductionof an augmented input space in PCE leads to a new uncertainty propagation algorithm for p-boxes. Theproposed algorithm is illustrated with two applications: a benchmark analytical function and a realistictruss structure. The results show that the proposed algorithm is capable of predicting the p-box of theresponse quantity extremely efficiently compared to double-loop Monte Carlo simulation.1. INTRODUCTIONIn modern engineering sciences, computationalsimulations, such as finite element modelling, havebecome wide spread. The goal is to predict the re-sponse of a system with respect to a set of param-eters, e.g. the deflection of a beam under variableloads. The parameters (e.g. geometries, mechanicalproperties, loads) are mapped to the quantity of in-terest through a computational model, e.g. throughthe governing equations of the process.It is only in recent times that the traditionallydeterministic model parameters have been gradu-ally substituted with probability distributions thataccount for their uncertainty. In practice though,data available for calibrating such distributions areoften too sparse, thus resulting in an extra layer ofuncertainty in their parameters. Different frame-works have been proposed to quantify the latterlack of knowledge (epistemic uncertainty) as wellas the natural variability of the process (aleatoryuncertainty), including probability-boxes (Fersonand Ginzburg, 1996), Bayesian hierarchical mod-els (Gelman, 2006) and Dempter-Shafer’s evidencetheory (Dempster, 1967; Shafer, 1976). Theseframeworks are generally referred to as impreciseprobabilities.After the input uncertainty is characterized,it must be propagated through a computationalmodel. The latter, however, is often an expensive-to-evaluate function, which can be replaced by anapproximate model, i.e. a meta-model, to reducethe computational effort needed. Well-known meta-modelling techniques include Polynomial ChaosExpansions (Ghanem and Spanos, 2003), Gaussianprocess modelling (a.k.a. Kriging) (Santner et al.,2003) and support vector machines (Gunn, 1998).This paper describes one formulation of impre-cise probabilities in Section 2 followed by an in-troduction to Polynomial Chaos expansions in Sec-tion 3. Finally these two ingredients are combined112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015in Section 4 and two applications are discussed inSection 5.2. INPUT UNCERTAINTY2.1. Probability theoryTraditionally, uncertainty in engineering has beentreated with probability theory.Consider a probability space (Ω,F ,P), where Ωdenotes the event space equipped with σ -algebraF and probability measure P. Random variablesare denoted by capital letters X(ω) : Ω→ DX ⊂ Rwhere ω ∈ Ω. A realization of variable X is de-noted by the corresponding lower case letters, e.g.x. Several random variables compose a randomvector X = [X1, . . . ,XM]T and the corresponding re-alizations x = [x1, . . . ,xM]T.In this context a random variable X is describedby its cumulative distribution function (CDF) FXwhich expresses the probability that X < x, i.e.FX(x) =P(X < x). The CDF has the properties thatit is monotone non-decreasing, that it tends to zerofor low values of x (FX(x→ −∞) = 0) and that ittends to one for high values of x (FX(x→ ∞) = 1).Note that such properties are valid for continuousas well as discrete random variables.For continuous random variables, the derivativeof a CDF is the probability density function fX(x) =dFX(x)/dx. The PDF describes the likelihood thatX is in the neighbourhood of x. Due to the factthat the CDF is non-decreasing, the PDF has non-negative values for all x ∈ X .As seen in the definitions above, probability the-ory offers a single measure (i.e. the probabilitymeasure) to describe variability in variable X . Inother words, we assume that the variability in X isknown and quantifiable by the CDF FX and the cor-responding PDF fX . This describes the case wherevariability is treated as the only source of uncer-tainty.2.2. Probability-boxA more general formulation is given by the frame-work of probability-boxes (p-boxes) which definesthe CDF of a variable X by its lower and upperbound distributions (Ferson and Ginzburg, 1996;Ferson and Hajagos, 2004). The idea is that due alack of knowledge (epistemic uncertainty), the CDFcannot be given a precise formulation. Thus theprobability-box framework accounts for aleatory aswell as for epistemic uncertainty in the descriptionof a variable X .The lower and upper boundaries of the CDF aredenoted by [FX ,FX ]. The true, but unknown, CDFof X lies within the boundaries for any value ofx ∈ X , i.e. FX(x) ≤ FX(x) ≤ FX(x), ∀x ∈ X . Theboundaries [FX ,FX ] mark the extreme cases of FXand are thus also CDFs by definition.The two boundaries form an intermediate spacein the variable-CDF-graph which resembles a box(see Figure 1), hence the name probability-box.−3 −2 −1 0 1 2 300. X BeliefPlausibilityP−boxFigure 1: P-box – CDF of a Gaussian variable withinterval-valued µX and σXThe p-box can be interpreted in the frameworkof Dempster-Shafer’s evidence theory (Dempster,1967; Shafer, 1976). The lower boundary FX de-scribes the minimum amount of probability thatmust be assigned to FX(x), which corresponds to thebelief function Bel(FX(x)) in the vocabulary of ev-idence theory. Analogously, the upper boundary ofthe p-box is associated with the maximum amountof probability that might be assigned to FX(x), orthe plausibility function Pl(FX(x)).Note that if FX(x) = FX(x) = FX(x), ∀x ∈ X ,then the p-box is called thin and conventional prob-ability theory can be applied.2.3. Parametric p-boxesIn the literature two types of p-boxes are identified,namely the free p-box and the parametric p-box. Inthis paper, we focus on parametric p-boxes (alsocalled distributional p-boxes).212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015A parametric p-box requires knowledge aboutthe shape of the true CDF but allows for uncertaintyin its parameters. The p-box is represented by afamily of distribution functions whose parametersθ lie within an interval. For a single variable X :FX(x) = FX(x,θ ), (1)where θi ∈ [θ i,θ i], i = 1, . . . ,nθ . This constructionresembles a Bayesian hierarchical model (Gelman,2006) in which the distribution of the parametersθ is replaced by an interval. This framework al-lows for a clear separation of aleatory and epistemicuncertainty: aleatory uncertainty is represented bythe distribution function family and epistemic un-certainty is represented by the interval on parame-ters θ .Figure 2 illustrates a parametric p-box generatedby a Gaussian random variable with mean valueand standard deviation varying within the intervalsµX = [−0.5,0.5] and σX = [0.7,1.0]. Several re-alizations of the CDF are shown. Note that in thecase of parametric p-boxes in general, lower/upperboundaries of the p-box are composed of several re-alizations of the p-box.−3 −2 −1 0 1 2 300. X  P−boxµ = 0, σ = 1µ = −0.5, σ = 0.7µ = 0.25, σ = 1Figure 2: Boundaries of a parametric p-box and somerealizations for specific parameter values θ ∗3. META-MODELLING3.1. Computational modelA computational model M is defined as a mappingof the M-dimensional input vector x to the outputscalar y, i.e. M : x ∈ DX ⊂ RM → y ∈ R. Due touncertainties in the input vector, the latter is repre-sented by the random vector X with joint CDF FX .The components of X = [X1, . . . ,XM]T are assumedindependent for the sake of simplicity throughoutthis paper. The model response is a random vari-able Y obtained by propagating the input randomvector X through the computational model M .Several techniques are available for surrogatingthe expensive-to-evaluate computational modelM .In the following section, Polynomial Chaos Expan-sions (Ghanem and Spanos, 2003; Sudret, 2007)will be briefly introduced.3.2. Polynomial Chaos ExpansionA well-known non-intrusive meta-modellingmethod is Polynomial Chaos Expansion (PCE)which approximates the computational model Mwith a finite series of polynomials orthogonal withrespect to the distribution of the input variables:Y ≈M (PCE)(X ) = ∑α∈A M,paαψα (X ), (2)where {aα ∈R} are the polynomial coefficients forthe multi-indices α = [α1, . . . ,αM] in the truncationset A M,p, M is input dimension, p is the maximumpolynomial degree and ψα (X ) are multivariate or-thonormal polynomials. Since the components of Xare assumed independent, the joint PDF is the prod-uct of the margins. For each marginal distributionfXi a functional inner product is defined:〈φ1,φ2〉i =∫Diφ1(x)φ2(x) fXi(x)dx. (3)For each input variable i = 1, . . . ,M a family of or-thonormal polynomials can be built that satisfies:〈ψ(i)j ,ψ(i)k 〉=∫Diψ(i)j (x)ψ(i)k (x) fXi(x)dx = δ jk,(4)where δ jk is the Kronecker symbol which is δ jk = 1for j = k and δ jk = 0 otherwise. A compilation ofcommon orthonormal univariate polynomials canbe found e.g. in Sudret (2014).3.3. Sparse PCEOne strategy to compute efficiently the coefficientsaα in Eq. (2) is linear regression, as introduced byBerveiller et al. (2006). Consider a set of N samplesof the input vector X = {χ(1), . . . ,χ(N)}, known312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015as the experimental design, and the correspond-ing responses of the exact computational modelY ={Y (1) =M (χ(1)), . . . ,Y (N) =M (χ(N))}.The set of coefficients aα can be computed throughthe solution of the least squares problem:aˆ = argmina∈R|A |1NN∑i=1(Y (i)− ∑α∈A M,paα ψα (χ(i)))2.(5)The efficiency of meta-modelling algorithms de-pends greatly on the choice of the set of polyno-mials A M,p (see Eq. (5)). Different strategies forlimiting the number of polynomials have been pro-posed including hyperbolic index sets (Blatman andSudret, 2011) which limit the total degree of poly-nomials and interactions. In case of high dimen-sionality (M ↑) this truncation scheme is not effi-cient enough to accurately estimate the model re-sponse Y and at the same time have a small numberof elements in A M,p.For this reason, algorithms have been developedto select out of a candidate set the polynomialsthat are most influential to the system response Y .Following Efron et al. (2004), Blatman and Sudret(2010) introduced the least angle regression selec-tion (LARS) algorithm for this purpose. LARS de-termines the sparse set of polynomials (out of acandidate set) that best describes the behaviour ofthe exact computational model based on the exper-imental design.4. UNCERTAINTY PROPAGATION OF P-BOXES4.1. Monte-Carlo-based propagationThe distinction of aleatory and epistemic uncer-tainty in the formulation of the parametric p-box al-lows one to propagate them separately. A straight-forward algorithm is the nested Monte Carlo al-gorithm (Eldred and Swiler, 2009; Chowdhary andDupuis, 2013) shown in Figure 3. In the outer loop,parameters of the CDF are sampled, i.e. θ (i) ∈ΘX .In the inner loop, a Monte Carlo simulation is con-ducted for estimating the CDF of the response valueY for a given input distribution FX (x,θ ). The set ofCDFs resulting from different values of θ (i) are fi-nally combined into a p-box. The boundaries of thep-box are obtained by:FY (y) = mini(FY (y,θ (i))), ∀y ∈DY (6)FY (y) = maxi(FY (y,θ (i))), ∀y ∈DY . (7)The nested Monte Carlo approach requires alarge number of model evaluations to accuratelypredict the p-box of the output variable Y . Thusthe algorithm becomes inefficient when the cost forevaluating the computational model M becomeslarge. Therefore we propose an algorithm to replacethe computational model M by its inexpensive-to-evaluate PCE surrogate.4.2. PCE-based p-box propagation4.2.1. Augmented input spaceConsider the parametric p-box from Section 2.3,which separates aleatory and epistemic uncertainty.The response of the computational model Y canbe interpreted as a function of the augmented in-put vector Zdef= [X ,ΘX ]T, where ΘX describes thespace of all parameters of all marginal distributions,e.g. ∏Mi=1[θ i,θ i], if the p-box of each Xi depends ona single parameter θi. The augmented input spaceleads to a PCE of dimension MZ = M+ |ΘX | where|ΘX | is the number of parameters:Y=M (aug)(X ,ΘX ) =M (aug)(Z). (8)Note that the parameters {θi, i = 1, . . . ,nθ} aregiven within interval boundaries [θ i,θ i] and aretreated in the PC expansion framework as a uni-formly distributed random variable within theseboundaries.PCE is defined on independent random variablesin the input space of the computational model,which is clearly not the case for parametric p-boxes because X is depending on the parametersΘX . Thus an isoprobabilistic transform (e.g. Nataftransform) of the augmented input space is requiredbefore calibrating the meta-model (Blatman andSudret, 2010).For illustration purposes, the case of a Gaussiandistribution is shown in this paper. Consider a para-metric p-box of a Gaussian random variable X withunknown mean value µX and standard deviation412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Inputparametricp-box FX,ΘOuter loopsampleparameterof the CDFθ(i) ∈ [θ,θ]Inner loopsampleconditionalmarginalsFX(x|θ(i))xj M(xj)j ← j + 1i ← i + 1Outputp-box[FY , FY ]Figure 3: Nested Monte Carlo approach – Propagation of imprecise probabilities by sampling the parameters θ(outer loop) and the input vectors x ∼ FX (x,θ) (inner loop)σX , denoted by X ∼ N ([µX ,µX ], [σX ,σX ]). Theisoprobabilistic transform T reads:X = µX +ξ ·σX =U1 +U2 ·U3, (9)where U1 =U (µX ,µX) and U2 =U (σX ,σX) de-note uniform random variables between the two ar-guments and ξ def= U3 =N (0,1) is a standard nor-mal random variable.The computational model can then be formulatedas a function of three independent random vari-ables:Y =M (X) =M (T (U1,U2,U3)) , (10)where T is shown in Eq. (9).Finally, the nested Monte Carlo algorithm de-scribed in Section 4.1 can be applied by substitutingthe full computational model M (X ) with its surro-gate in the augmented space M (aug)PCE (Z).4.2.2. Phantom pointsEq. (9) leads to an interesting feature of the experi-mental design in the augmented space. The compu-tational modelM is a function of |X |=M variableswhereas the augmented input space has MZ > Minput variables. Hence, for a given x ∈ DX thereare several combinations of {u1,u2,u3} such thatu1 +u2 ·u3 = x. This feature can be exploited whengenerating the experimental design of PC expan-sion models in the augmented space.Consider input variable X j and the associ-ated j-th component of the experimental de-sign, i.e. X j = {χ(1)j , . . . ,χ(N)j }. Each χ(k)j ∈DX j is a realization of the Normal distributionχ(k)j ∼ N(u(k)1, j,u(k)2, j). It can then be described asa function of the variables in the augmented inputspace, i.e. χ(k)j = T(u(k)1, j,u(k)2, j,u(k)3, j). Eq. (9) indi-cates that for Gaussian variables it holds:u(k)3, j =χ(k)j −u(k)1, ju(k)2, j. (11)Thus for each sample χ(k)j , u(k)3, j can be computed asa function of {u(k)1, j,u(k)2, j}.We define phantom points in the augmented in-put space as points which are obtained by sampling{u(k)1, j,u(k)2, j} and computing u(k)3, j by Eq. (11) resultingin the vector u(k)(i)j = {u(k)(i)1, j ,u(k)(i)2, j ,u(k)(i)3, j }, wherei = 1, . . .nph. Combining the j = 1, . . . ,M dimen-sions for the sample χ(k) leads to a maximum num-ber Nph = nMph phantom samples in the augmentedinput space. The entire experimental design hasthen a size of N×nMph samples.The key feature of the phantom points is that theyall correspond to the same χ(k) in the original space,with associated model response Y (k) =M (χ(k)).In other words, a single run of the model M yieldsup to nMph points in the augmented space.An infinite number of phantom samples could be512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015generated in principle. In practice however, onlya limited number is beneficial (see Section 5 andFigure 6).5. APPLICATION5.1. Rosenbrock functionThe Rosenbrock function is a two-dimensional,smooth, polynomial function defined as (Rosen-brock, 1960):M (x1,x2) = 100(x2− x21)2 +(1− x1)2. (12)The uncertainty associated with the two input vari-ables {x1,x2} is modelled by Gaussian randomvariables with interval-valued mean and standarddeviation. For both variables, µXi = [−0.5,0.5] andσXi = [0.7,1.0]. Figure 2 shows the p-box of theinput random variables as the region enclosed be-tween solid lines.The p-box of the response Y is obtained by apply-ing the algorithm in Section 4. The p-box is inter-preted as a parametric p-box for the nested MonteCarlo algorithm with and without meta-modelling(Section 4.2 and Section 4.1 respectively).The experimental design consists of N = 30Latin-hypercube samples generated from the para-metric p-box described in Section 2.3. The impre-cise parameters ΘX are interpreted as uniform ran-dom variables in order to cover the interval-valuedθ ∈ ΘX evenly. In the augmented space, Nph = 30phantom samples are used for each vector of theexperimental design χ(k) ∈ X , leading to a totalnumber of samples in the experimental design ofNtot =N ·Nph = 900 to build up the surrogate modelin the augmented space of dimension MZ = 6.The Nph phantom points for χ(k) areobtained as follows: Assuming thatz(k) = [u(k)1,1, u(k)2,1, u(k)3,1, u(k)1,2, u(k)2,2, u(k)3,2]T. ThroughLatin-hypercube sampling Nph samples are gen-erated in the m-dimensional ΘX space, whichdefines the components {[u(k)1,i ,u(k)2,i ], i = 1,2}.Then components [u(k)3,1,u(k)3,2] are computed fromEq. (11).The resulting p-boxes are shown in Figure 4 forboth algorithms. Solid lines and the grey arearepresent the exact p-box of the output variableY (double loop Monte Carlo simulation) whereasdiamonds represent the p-box from the surrogatemodel. Note that despite the small experimental de-sign the exact response p-box and the meta-modelbased p-box match perfectly. This behaviour wasexpected since the computational modelM and theisoprobabilistic transform T are polynomial func-tions.0 5000 10000 1500000. Y  Meta−modelExact modelFigure 4: Rosenbrock function – comparing the result-ing p-boxes for the exact model and the meta-model5.2. Linear elastic trussConsider the simply supported, linear-elastic trusspresented in Hurtado (2013) and sketched in Fig-ure 5. The computational model is a finite elementmodel of a this truss structure. The Young’s mod-ulus of all bars is E = 200 · 109 Pa whereasthe cross section of the bars varies: 0.00535 m2for the bars marked by •, 0.0068 m2 for the barsmarked by ◦ and 0.004 m2 for the remainingbars. The uncertainty in the input originates inthe seven loads {Pi, i = 1, . . . ,7} which are mod-elled as independent lognormal variables with meanvalue µPi = [95,105] kN and standard deviationσPi = [13,17] kN (Hurtado, 2013). The quantityof interest is the deflection at midspan denoted byu4 in Figure 5 as a function of the seven loads Pi.An experimental design of N = 100 Latin-hypercube samples following a parametric p-boxand varying number of phantom points Nph is gen-erated in a similar fashion as in Section 5.1. The to-tal number of samples in the experimental design inthe augmented input space is then Ntot = N ·Nph =100 ·Nph.Note that due to the lognormal distributions,Eq. (9) transforms into a function of the log-mean612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20158 x 2m 2mP1 P2 P3 P4 P5 P6 P7u4Figure 5: Truss structure – sketch of the geometry in-cluding the seven imprecise loads Pi and the targetdeflection u4λ and the log-standard deviation ζ for each load P(index i = 1, . . . ,7 has been omitted for clarity):P = exp [λ (U1,U2)+ζ (U1,U2) ·U3] , (13)where ζ (U1,U2) =√ln(1+(U2/U1)2),λ (U1,U2) = ln(U1)− ζ 2/2, U1 = U (95,105) kN,U2 = U (13,17) kN and U3 = N (0,1). Hence,Eq. (11) transforms into:u(k)3, j =ln(p(k)j)−λ (u(k)1, j,u(k)2, j)ζ (u(k)1, j,u(k)2, j). (14)Respecting the fact that the computational modelfor the beam deflection is a monotone function ofthe loads Pi, sampling the boundaries of the rectan-gular area defined by the ranges in {µPi,σPi} leadsto the boundaries of the p-box of the output variableu4.Figure 6 shows the boundaries of the p-box of thedeflection variable u4 for nMC,1 = 103 samples inthe outer loop and nMC,2 = 105 samples in the innerloop of the nested Monte Carlo algorithm using thePC expansion in the augmented space. Positive val-ues of u4 correspond to a deflection direction indi-cated in Figure 5. The different line styles representthe number of phantom points (Nph = {1,2,5,10}).The reference p-box of the response u4 is marked bydiamonds which display a nested MC algorithm us-ing the original finite element model with 103×105runs.The influence of the phantom points is clearlyvisible, since the p-boxes converge to a stable solu-tion for an increasing number of phantom points. Inthis case stable solutions are obtained with Nph > 4.0.02 0.022 0.024 0.026 0.028 0.03 0.03200. [m]F u 4  Nph = 1Nph = 2Nph = 5,10ReferenceFigure 6: Truss structure – resulting p-boxesNote that when conducting a reliability anal-ysis, the failure probability is given within therange provided by lower For instance u4,adm =0.028 m leads to a failure probability range of Pf =P(u4 ≥ u4,adm)= [2.3 ·10−3,1.8 ·10−1]. Note thatthese results are obtained using only 100 finite ele-ment runs.6. CONCLUSIONSThis paper deals with the propagation of uncer-tainty in the input of a computational model simu-lating a physical process. Due to sparsity of propercalibration data, the input parameters are modelledas imprecise probabilities, i.e. a combination ofaleatory and epistemic uncertainties. This is a typi-cal case in practice, where resources for generatingdata (i.e. measurements) are limited.One way to capture this lack of knowledge areprobability-boxes. Given parametric probability-boxes in the input variables, we propose an algo-rithm to propagate input uncertainty with the helpof Polynomial Chaos Expansions. The use of para-metric p-boxes allows for the separation of aleatoryand epistemic uncertainty in the meta-model by in-troducing an augmented input space. Such separa-tion is preserved in the p-box of the output variableof the system.An essential part of the algorithm are phan-tom points which are artificial experimental designpoints generated in the augmented input space with-out the need of additional expensive exact modelevaluations. They improve the accuracy of themeta-model without affecting computational re-712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015sources. This behaviour is due to the high redun-dancy of the augmented space formulation intro-duced to connect the p-boxes with PCE.The capabilities of the proposed algorithm areshown on two examples: a benchmark analyticalfunction and a more realistic engineering problem.In both cases the proposed algorithm is capableof predicting the response variable accurately withonly a small number of exact computational modelruns. This is of significance in practice where time,financial and computational resources are typicallylimited.Further studies will include modifications of theproposed algorithm to accurately estimate smallfailure probabilities for which Monte Carlo simu-lation is not efficient. This will include the use ofadaptive sampling algorithms for enriching experi-mental design continuously.7. REFERENCESBerveiller, M., Sudret, B., and Lemaire, M. (2006).“Stochastic finite elements: a non intrusive approachby regression.” Eur. J. Comput. Mech., 15(1-3), 81–92.Blatman, G. and Sudret, B. 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(2009). “Efficient algo-rithms for mixed aleatory-epistemic uncertainty quan-tification with application to radiation-hardened elec-tronics part I : algorithms and benchmark results.” Re-port No. SAND2009-5805, Sandia National Laborato-ries.Ferson, S. and Ginzburg, L. R. (1996). “Different meth-ods are needed to propagate ignorance and variabil-ity.” Reliab. Eng. Sys. Safety, 54(2-3), 133–144.Ferson, S. and Hajagos, J. G. (2004). “Arithmetic withuncertain numbers: rigorous and (often) best possibleanswers.” Reliab. Eng. Sys. Safety, 85(1-3), 135–152.Gelman, A. (2006). “Prior distributions for variance pa-rameters in hierarchical models (comment on articleby Browne and Draper).” Bayesian Anal., 1(3), 515–534.Ghanem, R. and Spanos, P. (2003). Stochastic FiniteElements : A Spectral Approach. Courier Dover Pub-lications.Gunn, S. (1998). “Support vector machines for classi-fication and regression.” Report No. ISIS-1-98, Dpt.of Electronics and Computer Science, University ofSouthampton.Hurtado, J. E. (2013). “Assessment of reliability inter-vals under input distributions with uncertain parame-ters.” Prob. Eng. Mech., 32, 80–92.Rosenbrock, H. (1960). “An automatic method for find-ing the greatest or least value of a function.” Comput.J., 3, 175–184.Santner, T. J., Williams, B. J., and Notz, W. I. (2003).The Design and Analysis of Computer Experiments.Springer, New York.Shafer, G. (1976). A mathematical theory of evidence.Princeton University Press, Princeton, NJ.Sudret, B. (2007). Uncertainty propagation and sensitiv-ity analysis in mechanical models – Contributions tostructural reliability and stochastic spectral methods.Université Blaise Pascal, Clermont-Ferrand, France.Habilitation à diriger des recherches, 173 pages.Sudret, B. 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