Propagation of Uncertainties Modelled by Parametric P-Boxes Using Sparse Polynomial Chaos Expansions Roland Schöbi, Bruno Sudret

Propagation of Uncertainties Modelled by Parametric P-boxes Using Sparse Polynomial Chaos Expansions Roland Schöbi, Bruno Sudret

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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

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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 behaviour of physical systems and processes. At the same time, awareness is growing on concepts of structural 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 (Ω,F ,P), where Ω x X, i.e. F (x) FX (x) FX (x), x X. The ∈ X ≤ ≤ ∀ ∈ denotes the event space equipped with σ-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(ω) : Ω DX R The two boundaries form an intermediate space → ⊂ where ω Ω. A realization of variable X is de- in the variable-CDF-graph which resembles a box ∈ 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 µX and σX 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- ∈ 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 conventional∀ prob-∈ 2.2. Probability-box ability theory can be applied. A more general formulation is given by the framework 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).

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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 θ 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,θ ), (1) X X the expensive-to-evaluate computational model M . In the following section, Polynomial Chaos Expan- where θi [θ i,θ i], i = 1,...,nθ . This construction resembles∈ a Bayesian hierarchical model (Gelman, sions (Ghanem and Spanos, 2003; Sudret, 2007) 2006) in which the distribution of the parameters will be briefly introduced. θ 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 θ . 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 ) = aα ψα (X ), (2) and standard deviation varying within the intervals ≈ ∑ α A M,p ∈ µX = [ 0.5,0.5] and σX = [0.7,1.0]. Several re- − alizations of the CDF are shown. Note that in the where aα R are the polynomial coefficients for { ∈ } case of parametric p-boxes in general, lower/upper the multi-indices α = [α1,...,αM] in the truncation boundaries of the p-box are composed of several re- set A M,p, M is input dimension, p is the maximum alizations of the p-box. polynomial degree and ψα (X ) are multivariate orthonormal polynomials. Since the components of X 1 P−box are assumed independent, the joint PDF is the prod- µ = 0, σ = 1 uct of the margins. For each marginal distribution 0.8 µ = −0.5, σ = 0.7 µ = 0.25, σ = 1 fXi a functional inner product is defined: 0.6 Z

X x x f x dx (3) F φ1,φ2 i = φ1( )φ2( ) X ( ) . h i i 0.4 Di For each input variable i = 1,...,M a family of or- 0.2 thonormal polynomials can be built that satisﬁes:

0 Z −3 −2 −1 0 1 2 3 (i) (i) (i) (i) x ψ j ,ψk = ψ j (x)ψk (x) fXi (x)dx = δ jk, h i Di Figure 2: Boundaries of a parametric p-box and some (4) realizations for specific parameter values θ ∗ where δ jk is the Kronecker symbol which is δ jk = 1 for j = k and δ jk = 0 otherwise. A compilation of common orthonormal univariate polynomials can 3. META-MODELLING be found e.g. in Sudret(2014). 3.1. Computational model A computational model M is defined as a mapping 3.3. Sparse PCE of the M-dimensional input vector x to the output One strategy to compute efficiently the coefficients M scalar y, i.e. M : x DX R y R. Due to aα in Eq. (2) is linear regression, as introduced by uncertainties in the input∈ vector,⊂ the→ latter∈ is repre- Berveiller et al.(2006). Consider a set of N samples (1) (N) sented by the random vector X with joint CDF FX . of the input vector X = χ ,..., χ , known { } 3 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015

as the experimental design, and the correspond- p-box are obtained by: ing responses of the exact computational model n 1 1 N N o (i) Y = Y ( ) = M (χ( )),...,Y ( ) = M (χ( )) . FY (y) = min FY (y,θ ) , y DY (6) i ∀ ∈ The set of coefﬁcients aα can be computed through (i) FY (y) = max FY (y,θ ) , y DY . (7) the solution of the least squares problem: i ∀ ∈

N !2 The nested Monte Carlo approach requires a 1 (i) (i) aˆ = argmin ∑ Y ∑ aα ψα (χ ) . large number of model evaluations to accurately A N − M,p a R| | i=1 α A predict the p-box of the output variable Y. Thus ∈ ∈ (5) the algorithm becomes inefficient when the cost for The efficiency of meta-modelling algorithms de- evaluating the computational model M becomes pends greatly on the choice of the set of polyno- large. Therefore we propose an algorithm to replace M,p mials A (see Eq. (5)). Different strategies for the computational model M by its inexpensive-to- limiting the number of polynomials have been pro- evaluate PCE surrogate. posed including hyperbolic index sets (Blatman and Sudret, 2011) which limit the total degree of poly- 4.2. PCE-based p-box propagation nomials and interactions. In case of high dimen- 4.2.1. Augmented input space sionality (M ) this truncation scheme is not effi- Consider the parametric p-box from Section 2.3, ↑ cient enough to accurately estimate the model re- which separates aleatory and epistemic uncertainty. sponse Y and at the same time have a small number The response of the computational model Y can of elements in A M,p. be interpreted as a function of the augmented in- For this reason, algorithms have been developed def T put vector Z = [X ,ΘX ] , where ΘX describes the to select out of a candidate set the polynomials space of all parameters of all marginal distributions, that are most influential to the system response Y. M e.g. ∏ [θ i,θ i], if the p-box of each Xi depends on Following Efron et al.(2004), Blatman and Sudret i=1 a single parameter θi. The augmented input space (2010) introduced the least angle regression selec- leads to a PCE of dimension MZ = M + ΘX where tion (LARS) algorithm for this purpose. LARS de- | | ΘX is the number of parameters: termines the sparse set of polynomials (out of a | | candidate set) that best describes the behaviour of (aug) (aug) Y = M (X ,ΘX ) = M (Z). (8) the exact computational model based on the experimental design. Note that the parameters θi, i = 1,...,n are { θ } given within interval boundaries [θ i,θ i] and are 4. UNCERTAINTY PROPAGATION OF P- treated in the PC expansion framework as a uni- BOXES formly distributed random variable within these 4.1. Monte-Carlo-based propagation boundaries. The distinction of aleatory and epistemic uncer- PCE is defined on independent random variables tainty in the formulation of the parametric p-box al- in the input space of the computational model, lows one to propagate them separately. A straight- which is clearly not the case for parametric p- forward algorithm is the nested Monte Carlo al- boxes because X is depending on the parameters gorithm (Eldred and Swiler, 2009; Chowdhary and ΘX . Thus an isoprobabilistic transform (e.g. Nataf Dupuis, 2013) shown in Figure3. In the outer loop, transform) of the augmented input space is required (i) parameters of the CDF are sampled, i.e. θ ΘX . before calibrating the meta-model (Blatman and ∈ In the inner loop, a Monte Carlo simulation is con- Sudret, 2010). ducted for estimating the CDF of the response value For illustration purposes, the case of a Gaussian Y for a given input distribution FX (x,θ ). The set of distribution is shown in this paper. Consider a para- CDFs resulting from different values of θ (i) are fi- metric p-box of a Gaussian random variable X with nally combined into a p-box. The boundaries of the unknown mean value µX and standard deviation

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Outer loop Inner loop sample conditional xj (x ) sample marginals M j Input parameter θ(i) [θ, θ] (i) Output ∈ FX (x θ ) parametric of the CDF | p-box p-box FX,Θ [F Y , F Y ] j j + 1 ←

i i + 1 ←

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) ∼

, denoted by X . The Consider input variable X and the associ- σX N ([µX , µX ],[σ X ,σ X ]) j isoprobabilistic transform∼ T reads: ated j-th component of the experimental de- (1) (N) (k) sign, i.e. X j = χ ,..., χ . Each χ { j j } j ∈ X = µX + ξ σX = U1 +U2 U3, (9) is a realization of the Normal distribution · · DXj k k k χ( ) u( ),u( ) . It can then be described as where U and U de- j N 1, j 2, j 1 = U (µX , µX ) 2 = U (σ X ,σ X ) ∼ note uniform random variables between the two ar- a function of the variables in the augmented input def (k) (k) (k) (k) guments and ξ = U3 = N (0,1) is a standard nor- space, i.e. χ j = T u1, j,u2, j,u3, j . Eq. (9) indi- mal random variable. cates that for Gaussian variables it holds: The computational model can then be formulated (k) (k) as a function of three independent random vari- k χ j u1 j u( ) = − , . (11) ables: 3, j (k) u2, j

Y = M (X) = M (T(U1,U2,U3)), (10) (k) (k) Thus for each sample χ j , u3, j can be computed as (k) (k) where T is shown in Eq. (9). a function of u ,u . { 1, j 2, j} Finally, the nested Monte Carlo algorithm de- We deﬁne phantom points in the augmented in- scribed in Section 4.1 can be applied by substituting put space as points which are obtained by sampling (k) (k) (k) the full computational model M (X ) with its surro- u1, j,u2, j and computing u3, j by Eq. (11) resulting (aug) { } (k)(i) (k)(i) (k)(i) (k)(i) gate in the augmented space MPCE (Z). in the vector u = u ,u ,u , where j { 1, j 2, j 3, j } i = 1,...nph. Combining the j = 1,...,M dimen- 4.2.2. Phantom points sions for the sample χ(k) leads to a maximum num- M Eq. (9) leads to an interesting feature of the experi- ber Nph = nph phantom samples in the augmented mental design in the augmented space. The compu- input space. The entire experimental design has tational model M is a function of X = M variables then a size of N nM samples. | | × ph whereas the augmented input space has MZ > M The key feature of the phantom points is that they (k) input variables. Hence, for a given x DX there all correspond to the same χ in the original space, ∈ (k) (k) are several combinations of u1,u2,u3 such that with associated model response Y = M (χ ). { } u1 +u2 u3 = x. This feature can be exploited when In other words, a single run of the model M yields · M generating the experimental design of PC expan- up to nph points in the augmented space. sion models in the augmented space. An inﬁnite number of phantom samples could be

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generated in principle. In practice however, only diamonds represent the p-box from the surrogate a limited number is beneﬁcial (see Section5 and model. Note that despite the small experimental de- Figure6). sign the exact response p-box and the meta-model based p-box match perfectly. This behaviour was 5. APPLICATION expected since the computational model M and the 5.1. Rosenbrock function isoprobabilistic transform T are polynomial func- The Rosenbrock function is a two-dimensional, tions. smooth, polynomial function deﬁned as (Rosen- brock, 1960): 1

2 2 2 0.8 M (x1,x2) = 100(x2 x ) + (1 x1) . (12) − 1 − The uncertainty associated with the two input vari- 0.6 Y ables x ,x is modelled by Gaussian random F 1 2 0.4 variables{ with} interval-valued mean and standard

deviation. For both variables, µX = [ 0.5,0.5] and 0.2 i − σX = [0.7,1.0]. Figure2 shows the p-box of the Meta−model i Exact model 0 input random variables as the region enclosed be- 0 5000 10000 15000 tween solid lines. y The p-box of the response Y is obtained by apply- Figure 4: Rosenbrock function – comparing the resulting the algorithm in Section4. The p-box is inter- ing p-boxes for the exact model and the meta-model preted as a parametric p-box for the nested Monte Carlo algorithm with and without meta-modelling (Section 4.2 and Section 4.1 respectively). 5.2. Linear elastic truss The experimental design consists of N = 30 Consider the simply supported, linear-elastic truss Latin-hypercube samples generated from the para- presented in Hurtado(2013) and sketched in Fig- metric p-box described in Section 2.3. The impre- ure5. The computational model is a finite element cise parameters Θ are interpreted as uniform ran- model of a this truss structure. The Young’s mod- ΘX 9 dom variables in order to cover the interval-valued ulus of all bars is E = 200 10 Pa whereas · 2 θ Θ evenly. In the augmented space, N 30 the cross section of the bars varies: 0.00535 m θ ΘX ph = 2 phantom∈ samples are used for each vector of the for the bars marked by , 0.0068 m for the bars • 2 (k) marked by and 0.004 m for the remaining experimental design χ X , leading to a total ◦ number of samples in the∈ experimental design of bars. The uncertainty in the input originates in the seven loads Pi, i = 1,...,7 which are mod- Ntot = N Nph = 900 to build up the surrogate model · elled as independent{ lognormal variables} with mean in the augmented space of dimension MZ = 6. (k) value µP = [95,105] kN and standard deviation The Nph phantom points for χ are i obtained as follows: Assuming that σPi = [13,17] kN (Hurtado, 2013). The quantity k k k k k k of interest is the deflection at midspan denoted by z(k) = [u( ), u( ), u( ), u( ), u( ), u( )]T. Through 1,1 2,1 3,1 1,2 2,2 3,2 u in Figure5 as a function of the seven loads P. Latin-hypercube sampling N samples are gen- 4 i ph An experimental design of N = 100 Latin- erated in the m-dimensional ΘX space, which k k hypercube samples following a parametric p-box defines the components [u( ),u( )], i = 1,2 . 1,i 2,i and varying number of phantom points Nph is gen- (k) (k{) } Then components [u3,1,u3,2] are computed from erated in a similar fashion as in Section 5.1. The to- Eq. (11). tal number of samples in the experimental design in The resulting p-boxes are shown in Figure4 for the augmented input space is then Ntot = N Nph = · both algorithms. Solid lines and the grey area 100 Nph. represent the exact p-box of the output variable Note· that due to the lognormal distributions, Y (double loop Monte Carlo simulation) whereas Eq. (9) transforms into a function of the log-mean

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1 8 x 2m

0.8 2m P P P P P P P 0.6 4 u u F 0.4 N = 1 ph N = 2 Figure 5: Truss structure – sketch of the geometry in- 0.2 ph N = 5,10 cluding the seven imprecise loads P and the target ph i Reference 0 deﬂection u4 0.02 0.022 0.024 0.026 0.028 0.03 0.032 u [m] 4

λ and the log-standard deviation ζ for each load P Figure 6: Truss structure – resulting p-boxes (index i = 1,...,7 has been omitted for clarity): Note that when conducting a reliability anal- P = exp[λ(U1,U2) + ζ(U1,U2) U3], (13) · ysis, the failure probability is given within the p 2 range provided by lower For instance u4,adm = where ζ(U1,U2) = ln(1 + (U2/U1) ), 2 0.028 m leads to a failure probability range of Pf = λ(U1,U2) = ln(U1) ζ /2, U1 = U (95,105) kN, 3 1 − P u4 u4,adm = [2.3 10− ,1.8 10− ]. Note that U2 = U (13,17) kN and U3 = N (0,1). Hence, ≥ · · Eq. (11) transforms into: these results are obtained using only 100 ﬁnite element runs. k k k ln p( ) λ(u( ),u( )) (k) j − 1, j 2, j 6.CONCLUSIONS u3 j = . (14) , (k) (k) This paper deals with the propagation of uncer- ζ(u1, j,u2, j) tainty in the input of a computational model simu- Respecting the fact that the computational model lating a physical process. Due to sparsity of proper for the beam deﬂection is a monotone function of calibration data, the input parameters are modelled the loads Pi, sampling the boundaries of the rectan- as imprecise probabilities, i.e. a combination of

gular area defined by the ranges in µPi ,σPi leads aleatory and epistemic uncertainties. This is a typi- to the boundaries of the p-box of the{ output variable} cal case in practice, where resources for generating u4. data (i.e. measurements) are limited. Figure6 shows the boundaries of the p-box of the One way to capture this lack of knowledge are 3 deflection variable u4 for nMC,1 = 10 samples in probability-boxes. Given parametric probability- 5 the outer loop and nMC,2 = 10 samples in the inner boxes in the input variables, we propose an algo- loop of the nested Monte Carlo algorithm using the rithm to propagate input uncertainty with the help PC expansion in the augmented space. Positive val- of Polynomial Chaos Expansions. The use of para- ues of u4 correspond to a deflection direction indi- metric p-boxes allows for the separation of aleatory cated in Figure5. The different line styles represent and epistemic uncertainty in the meta-model by in- the number of phantom points (Nph = 1,2,5,10 ). troducing an augmented input space. Such separa- { } The reference p-box of the response u4 is marked by tion is preserved in the p-box of the output variable diamonds which display a nested MC algorithm us- of the system. ing the original finite element model with 103 105 An essential part of the algorithm are phan- runs. × tom points which are artificial experimental design The influence of the phantom points is clearly points generated in the augmented input space with- visible, since the p-boxes converge to a stable solu- out the need of additional expensive exact model tion for an increasing number of phantom points. In evaluations. They improve the accuracy of the this case stable solutions are obtained with Nph > 4. meta-model without affecting computational re-

7 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 sources. This behaviour is due to the high redun- ods are needed to propagate ignorance and variabil- dancy of the augmented space formulation intro- ity.” Reliab. Eng. Sys. Safety, 54(2-3), 133–144. duced to connect the p-boxes with PCE. Ferson, S. and Hajagos, J. G. (2004). “Arithmetic with The capabilities of the proposed algorithm are uncertain numbers: rigorous and (often) best possible shown on two examples: a benchmark analytical answers.” Reliab. Eng. Sys. Safety, 85(1-3), 135–152. function and a more realistic engineering problem. Gelman, A. (2006). “Prior distributions for variance parameters in hierarchical models (comment on article In both cases the proposed algorithm is capable by Browne and Draper).” Bayesian Anal., 1(3), 515– of predicting the response variable accurately with 534. only a small number of exact computational model Ghanem, R. and Spanos, P. (2003). Stochastic Finite runs. This is of significance in practice where time, Elements : A Spectral Approach. Courier Dover Pub- financial and computational resources are typically lications. limited. Gunn, S. (1998). “Support vector machines for classi- Further studies will include modifications of the fication and regression.” Report No. ISIS-1-98, Dpt. proposed algorithm to accurately estimate small of Electronics and Computer Science, University of failure probabilities for which Monte Carlo simu- Southampton. lation is not efficient. This will include the use of Hurtado, J. E. (2013). “Assessment of reliability inter- adaptive sampling algorithms for enriching experi- vals under input distributions with uncertain parame- mental design continuously. 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. 7.REFERENCES J., 3, 175–184. Berveiller, M., Sudret, B., and Lemaire, M. (2006). Santner, T. J., Williams, B. J., and Notz, W. I. (2003). “Stochastic finite elements: a non intrusive approach The Design and Analysis of Computer Experiments. by regression.” Eur. J. Comput. Mech., 15(1-3), 81– Springer, New York. 92. Shafer, G. (1976). A mathematical theory of evidence. Blatman, G. and Sudret, B. (2010). “An adaptive al- Princeton University Press, Princeton, NJ. gorithm to build up sparse polynomial chaos expan- Sudret, B. (2007). Uncertainty propagation and sensitiv- sions for stochastic finite element analysis.” Prob. ity analysis in mechanical models – Contributions to Eng. Mech., 25(2), 183–197. structural reliability and stochastic spectral methods. Blatman, G. and Sudret, B. (2011). “Adaptive sparse Université Blaise Pascal, Clermont-Ferrand, France. polynomial chaos expansion based on Least Angle Habilitation à diriger des recherches, 173 pages. Regression.” J. Comput. Phys, 230, 2345–2367. Sudret, B. (2014). “Polynomial chaos expansions Chowdhary, K. and Dupuis, P. (2013). “Distinguish- and stochastic finite element methods.” Risk Reliab. ing and integrating aleatoric and epistemic variation Geotech. Eng., K. K. Phoon and J. Ching, eds., Tay- in uncertainty quantification.” ESAIM Math. Model. lor and Francis, Chapter 6. Numer. Anal., 47(3), 635–662. Dempster, A. P. (1967). “Upper and lower probabilities induced by multivalued mapping.” Ann. Math. Stat., 38(2), 325–339. Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004). “Least angle regression.” Ann. Stat., 32, 407– 499. Eldred, M. S. and Swiler, L. P. (2009). “Efficient algorithms for mixed aleatory-epistemic uncertainty quantification with application to radiation-hardened electronics part I : algorithms and benchmark results.” Re- port No. SAND2009-5805, Sandia National Laborato- ries. Ferson, S. and Ginzburg, L. R. (1996). “Different meth-