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

Compressive polynomial chaos expansion for multidimensional model maps Marelli, Stefano; 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, 2015Compressive Polynomial Chaos Expansion for Multi DimensionalModel MapsStefano MarelliSenior Assistant, Chair of Risk, Safety and Uncertainty ETH Zürich, SwitzerlandBruno SudretProfessor, Chair of Risk, Safety and Uncertainty Quantification, ETH Zürich, SwitzerlandABSTRACT: Modern high-resolution numerical models used in engineering often produce multi-dimensional maps of outputs (e.g. nodal displacements on a FEM mesh) that may result in more than 105highly correlated outputs for each set of model parameters. Most available metamodelling techniques,however, are not yet suitable for handling such large maps, including Polynomial Chaos Expansions(PCE). Indeed, the PCE of a numerical model with many outputs is traditionally handled by indepen-dently metamodelling each one of them. We introduce a two-stage PCE approach that aims at solvingthis problem: in the first stage, PCE is used to compress the map of outputs on a much sparser basis in themap coordinates; in the second stage, standard PCE of the compressed map is carried out w.r.t. the un-derlying model parameters. Standard PCE post-processing techniques are then used to derive analyticalexpressions for several stochastic properties of the resulting compressive PCE.1. INTRODUCTIONPolynomial Chaos Expansions (PCE) are a well es-tablished tool in Uncertainty Quantification (UQ),in both applied mathematics and engineering ap-plications. Extensive literature is available on ef-ficient non-intrusive techniques for calculating they j coefficients, such as spectral projection andleast-square regression (see e.g., Sudret (2007)and references therein). However, a field of re-search that remains relatively unexplored is that ofnon-intrusive metamodelling of multi-dimensionalmodel maps. Examples of such maps include time-dependent solutions of partial differential equa-tions (PDE) and force/displacement fields calcu-lated by finite element modelling FEM on multi-dimensional meshes. In their seminal work in thefield of metamodelling of vector-valued functions,Blatman and Sudret (2013) proposed compressingthe model response map via principal componentanalysis (PCA) and metamodelling each significantprincipal component independently. We herein pro-pose an extension of this approach that uses PCE asthe compression tool to represent complex (but reg-ular) high-dimensional model maps efficiently. Dueto the linearity of PCE, point-by-point convergencein the full-model space ensures similar convergencein the compressed model response.In Section 2, we give a brief introduction to PCEfor scalar and vector-valued functions. In Section 3,we introduce the concept of model maps and theformalism of compressive polynomial chaos expan-sions (CPCE). We also provide an algorithm for thenon-intrusive calculation of the CPCE coefficients.In Section 4, we discuss and derive analytical ex-pressions for several stochastic properties of CPCE.We then apply the CPCE on a test 2D function inSection 5. Finally, we summarize and give conclud-ing remarks in Section 6.2. POLYNOMIAL CHAOS EXPANSIONS2.1. Basic definitionsPolynomial Chaos Expansion (PCE) is a spec-tral decomposition technique that allows one torepresent a finite-variance scalar-output functionY = M (Ξ) as:Y =M (Ξ) =∞∑j=0a jΨ j(Ξ) (1)112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015where Ξ ∈ RM is a random vector of uncertain pa-rameters, a j ∈R is a set of scalar coefficients (spec-tral coordinates) and the Ψ j(Ξ)∈R form a polyno-mial orthonormal basis w.r.t. the functional scalarproduct:〈g(ξ ),h(ξ )〉=∫DΞg(ξ )h(ξ ) fΞ(ξ )dξ (2)whereDΞ is the support of Ξ and fΞ(ξ ) is a positivesemi-definite weight function that satisfies:∫DΞfΞ(ξ )dξ = 1. (3)Note that, due to its properties, it is customary tointerpret fΞ(ξ ) as the probability density function(PDF) of the random variable Ξ in stochastic PCEapplications. Nevertheless, the spectral decomposi-tion in Eq. (1) is, in a more general context, a func-tional approximation of the model function M (Ξ)as long as an appropriate weight function fΞ(ξ ) isused in the definition of the scalar product in Eq (2).The orthonormality condition on the basis ele-ments Ψ j reads:〈Ψi(Ξ),Ψ j(Ξ)〉= δi j. (4)where δi j is the Kronecker symbol.2.2. Sparsity of PCENote that the sum in Eq. (1) has an infinite num-ber of terms, hence requiring truncation strategiesfor computational purposes. In practice, however,smooth models tend to have quickly decaying PCEcoefficients, hence allowing for accurate basis trun-cation strategies to be devised. The truncated formof Eq. (1) reads:Y (Ξ)' ∑α∈A M,paαΨα (Ξ) (5)where α = {α1,α2, ...,αM} is a multi-index thatidentifies the polynomial degree of Ψα in each ofthe M input variables, while A M,p is a finite cardi-nality P=∣∣A M,p∣∣ set of multi-indices of dimensionM and maximum degree p.The cardinality P can be controlled in manyways, e.g. via maximum-polynomial degree trunca-tion strategies (e.g. hyperbolic truncation, see Blat-man and Sudret, 2010) or via sparsity-favouring co-efficient calculation strategies (e.g. through LeastAngle Regression, Blatman and Sudret, 2011).2.3. PCE of vector-valued modelsA trivial extension of Eq. (1) to the case of mod-els with vector-valued outputs Y ={Y 1,Y 2, ...,Y n}consists in separately expanding each of the outputsindependently on the same truncated basis:Y i (Ξ)' ∑α∈A M,paiαΨα (Ξ), i = 1, ...,n (6)Due to the orthonormality of the Ψ(ξ ) in Eq. (1)given in Eq. (4), it is trivial to demonstrate that themean value of each of the components of Y is givenby:µ iY = ai0 (7)where a0 represents the coefficient of the constantbasis term. Correspondingly, the shared polyno-mial basis ensures that the covariance informationbetween the elements of Y is also encoded in the co-efficients aiα , even though they are calculated inde-pendently for each outputs. In fact, the covariancematrix of the elements of Y can be written as:Ci jYdef= E[(Y i−µ iY )(Y j−µjY )](8)By substituting Eq. (6) and Eq. (7) into Eq. (8),one obtains (the reference to the random variable Ξis dropped for better readability):Ci jY = E[∑α 6=0aiαΨα ∑β 6=0a jβΨβ](9)where the notation α 6= 0 stands for α ∈A M,p\0 .Due to the orthonormality of the basis w.r.t. tothe expectation value in Eq (4), it immediately fol-lows that:Ci jY = ∑α 6=0aiαajα (10)3. COMPRESSIVE POLYNOMIAL CHAOS3.1. Model mapsIn many common modelling scenarios, the modelresponse to a sample of uncertain model parametersξ is not simply a scalar, but it is a d−dimensionalmap, e.g. the displacements at the nodes of a com-plex finite element model (FEM). Hence, we definea model map as:M (Ξ,X ) ∈ R (11)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015where Ξ ∈ RM is a random vector with probabil-ity density function fΞ(ξ ) that describes the uncer-tainty in the model parameters, while X ∈ Rd is aset of coordinates in the appropriate model space(e.g., Cartesian coordinates, or time). An exampleof model map with d = 1 could be the vertical dis-placement along the length of a simply supportedbeam with uncertain Young’s modulus. In this case,X ∈ R would represent the coordinate along thebeam length, while Ξ the uncertain Young’s modu-lus.Albeit strictly speaking not a random variable,a deterministic coordinate X on a bounded do-main can be seen as a random variable suitablydistributed on the entire map. In the case of aFEM model with a regular mesh, the node loca-tions are uniformly distributed throughout the do-main. Similarly, a finite-difference solution of atime-dependent PDE is typically calculated withregular time sampling.In this contribution, we consider each calcula-tion of a model map for a given realization ξ (i)of the input random vector Ξ as a sample from ascalar modelM (ξ (i),X ) with a d-dimensional ran-dom input vector X . A d = 2 example of two re-alizations of such a map is shown for reference inFigure 1.In typical applications, such sample can includeseveral tens up to hundreds thousands of values, asis often the case with high fidelity FEM models.Very high resolutions, however, are often used dueto numerical stability requirements, even in caseswhen the actual model response is very smooth.With no loss of generality, in this paper we will onlyconsider applications where the sampling in the co-ordinate space is uniform. Note that there are noconstraints in the regularity of the sampling, nor inthe number of samples, which may even vary be-tween different realizations of the random parame-ters ξ (i), as long as their distribution is uniform.3.2. Polynomial map compressionWhen metamodelling a function with a large num-ber of outputs, the classical approach of performingfull polynomial chaos expansion for each one of theoutputs can quickly become impractical. Due to theindependence of Ξ and X , for any given realizationξ of the input random vector Ξ, the truncated PCEin Eq. (5) of a model map realization on its coordi-nates can be written as follows:M (ξ ,X )' ∑γ∈A d,pCcγ (ξ )Φγ (X ) (12)with orthonormal polynomial basisΦγ (X ) =d∏i=1φγi(Xi). (13)Please note that, due to the independence betweenΞ and X , the coefficients cγ (ξ ) are just scalars inthe context of Eq. (12).If the model map M (Ξ,X ) is smooth in theX coordinates, the number of coefficients cγ (ξ )is typically limited to a few tens to several hun-dreds elements. Specifically, the number of coef-ficients equals the cardinality of the truncation setPC = |A d,pC |.Considering each of the cγ (ξ ) as a function onthe random input vector Ξ, we can further expandeach coefficient as:cγ (ξ )' ∑α∈A M,paγαΨα (ξ ) (14)which, when plugged back into Eq. (12) reads:M (ξ ,X )'∑γ∑αaγαΨα (ξ )Φγ (X ). (15)Equation (15) is the compressive PCE equation(CPCE). In case of a numerical smooth model mapwith a very large number of outputs for each real-ization of the random parameters ξ , the outputs arefirst compressed to their sparse PCE coefficients,which in turn are surrogated via PCE on the origi-nal input random variables.3.3. Non-intrusive calculation of the CPCE coef-ficientsPractically, the non-intrusive calculation of the co-efficients aγα in Eq. (14) and Eq. (15) requires atwo-step least-squares algorithm. For details aboutleast-squares-based calculation of PCE coefficients,with emphasis on sparsity, please refer to Blatmanand Sudret (2011).312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015X 1X 2  −1 −0.5 0 0.5 1−1−0.500.51−4−3−2−1012X 1X 2  −1 −0.5 0 0.5 1−1−0.500.5100.511.52Figure 1: Two realizations of the model map in Eq. (23) calculated on a regular grid with n = 105 nodes. Eachmap corresponds to one realization ξ (i) of the input random vector Ξ.In the first stage, an experimental designE ={ξ (1),ξ (2), ...,ξ (N)}is created as a suit-ably sized sample of the input random vector Ξ,and the corresponding model maps are calculatedY ={y(1),y(2), ...,y(N)}. Each y(i) is a vector ofscalar model responses at the corresponding set ofcoordinates X (i) ={x(i)(1),x(i)(2), ...,x(i)(n(i))}. Pleasenote that there is no restriction on the size n(i) ofeach model map realization, as long as all map real-izations are defined on the same domain and drawnfrom the same distribution. This property is impor-tant in practice, because many model maps have re-sponses defined on coordinate sets that depend onthe corresponding realization ξ (i) of the random pa-rameters Ξ. A typical example would be the finitedifference solution of a PDE in time, whose time-step (that defines the coordinates X (i)) depends onthe model parameters to ensure numerical stability.When the experimental design Y is avail-able, a set of compressive PCE coefficientsc(i) = {cγ 1(ξ(i)),cγ 2(ξ(i)), ...,cγPC (ξ(i))} is cal-culated according to Eq. (14) for each one of theexperimental design samples y(i). The coefficientsare then grouped as C ={c(1),c(2), ...,c(N)}. In atypical engineering scenario, M d and n(i) N,hence allowing the compression to achieve high ac-curacy in the coordinates X at a relatively low com-putational cost.In the second stage, the compressed experimentaldesign C is used to calculate the CPCE coefficientsaγα in Eq. (14).3.4. Considerations on convergenceDue to the linear nature of PCE, the compres-sion coefficients aγ (ξ ) share similar propertiesof smoothness and finite variance as the originalmodel map in equation Eq. (6). Hence, if the point-wise PCE on ξ of a model map is sparse and con-vergent for any x, so will be its compressive coun-terpart in Eq. (15).4. POST-PROCESSING OF CPCE4.1. Moments of a CPCEAn important property of PCE is that its coefficientsencode substantial information about the stochasticproperties (e.g. moments) of the model response,as shown in Section 2.3. Due to the separation be-tween physical coordinates and random input pa-rameters in Eq. (12), the model map at each coordi-nate point x can be considered as a linear superim-position of the random variables cγ (ξ ) with coeffi-cients Φ(x). It is therefore easy to extend Eqs. (7)and (10) through Eq. (14):µY (x) =∑γaγ0Φγ (x) (16)for the mean value at a point x. Correspondingly,the covariance between any two coordinate points(x,x′) reads:CY (x,x′) =∑γ∑γ ′Cγ γ′c Φγ (x)Φγ ′(x′) (17)where Cγ γ′c is the covariance matrix of the ran-dom coefficients cγ (ξ ) calculated according to412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Eq. (10). Note that this result is equivalent to build-ing a classical PCE in an augmented input spaceΞ(aug) = {Ξ,X}. Such an approach, however,would be computationally more challenging, as itwould require building a suitable PCE basis to ac-count for the separability of the model map w.r.t.Ξ and X , as well as solving a higher dimensionalleast squares minimization with a very large exper-imental design, instead of solving N independent,d-dimensional ones.4.2. Variance decomposition and Sobol’ indicesAs demonstrated in Sudret (2008), a close relationexists between variance decomposition, and PCEcoefficients. Such relation is particularly useful forthe calculation of the so-called Sobol’ global sensi-tivity indices, a well established tool in sensitivityanalysis.The basic form of variance decomposition can bewritten as follows (Sobol’ (2001)):f (ξ ) =M∑i=1fi(ξi)+∑i 6= jfi j(ξi,ξ j)+ ...+ f12...M(ξ1,ξ2, ...,ξM)+ f0(18)where the fi j...s are scalar functions depending onthe subset of input variables {ξi,ξ j, ...,ξs}.The coefficients in Eq. (14) can be grouped ac-cording to the functional dependence of the corre-sponding Ψα (Sudret, 2008):cγ(ξ ) = ∑α∈A M,piaγαΨα (ξi)+ ∑α∈A M,pi jaγαΨα (ξi,ξ j)+ ...+aγαΨα (ξi,ξ j, ...,ξM)+αγ0 (19)def= c{i}γ + c{i j}γ + ...+ c{12...M}γ + cγ0where the set of multi-indices:AM,pi j...s = {α ∈AM,p : αk > 0 ∀k ∈ {i, j, ...s} ,αl = 0 ∀l /∈ {i, j, ...s}} (20)identifies the basis elements that depend on the sub-set of the input variables {i, j, ...s}. Because of theuniqueness of the two representations in Eqs. (14)and (18), each sum in Eq. (18) can be identifiedwith the corresponding sum in Eq. (19).The Sobol’ indices can be defined as the ratio ofthe variance of each term in Eq. (18) Di j...s to thetotal variance D:Si j...s = Di j...s/D. (21)By calculating the covariance matrices Cγ γ′c{i j...s}ofthe sub-PCE compression coefficients c{i j...s}γ inEq. (19) with Eq. (10), it is trivial to derive the map-equivalent of the Sobol’ indices in Eq. (21) startingfrom the map-covariance in Eq. (17):Si j...s(x) =1CY (x,x)∑γ∑γ ′Cγ γ′c{i j...s}Φγ (x)2. (22)5. EXAMPLE APPLICATION: 2D MAP5.1. A 2D analytical mapTo validate the method proposed and the cor-responding post-processing properties, an ad-hoccomplex 2D analytical map is created according to:Y (ξ ,x) = e−12 (x1ξ1)+10ξ 21 ξ2x22+sin(−pi(x21 +ξ2x2)2)(23)The map coordinates span a rectangular regionx1,2 ∈ [−1,1]. Each model evaluation returns themap value on a regular grid of 400×250 (n = 105)points in the coordinate space. Some examplemodel maps for different realizations of Ξ areshown for reference in Figure 1.The input random vector is chosen asΞ ∼ U (−1,1)2. Albeit in this particularcase both the bounds and the dimensionality onΞ and X coincide, this condition is neither re-quired nor recommended. Indeed, in many typicalengineering scenarios, M d.The choice of such a seemingly complex modelstems from the need to meet the following criteriafor demonstration purposes:• the dimensionalities M and d must be lowenough to allow for effective visualization;• the model must be sufficiently complex andnot simply polynomial;512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015• the model must exhibit sufficient variancew.r.t. the random inputs Ξ;• the model must have predictable distributionsfor moments and sensitivity indices.5.2. The experimental designThe experimental design used in this analysis con-sists of N = 200 realizations of the input randomvector Ξ, hence resulting in a corresponding num-ber of model maps similar to those in Figure 1.Following the approach outlined in Section 3.3,the set of model responses Y is calculated on theexperimental design E ={ξ (1),ξ (2), ...,ξ (N)}.The corresponding set of compressive- ex-perimental designs in the map coordinatesX (i) ≡X ={x(1),x(2), ...,x(n)}is identical foreach experimental design sample ξ (i): the regulargrid with n = 105 nodes described in Section Experimental design compressionFor the calculation of the compressive coeffi-cients cγ in Eq. (14), the regular grid X istreated as a uniform sampling of a random vectorX ∼ U (−1,1)2. Ordinary least squares regres-sion with maximum polynomial degree pc = 25 ind = 2 dimensions is then performed. A hyperbolictruncation scheme with q = 0.7 is chosen to definethe PCE basis (Blatman and Sudret, 2011). Thechoice of pc and q is based on a preliminary analy-sis of a random subset of the available experimen-tal design. More generally, it is possible to applyany adaptive sparse PCE algorithms to determinethe smallest set of basis elements suitable to accu-rately represent all the maps in the experimental de-sign. The resulting estimated generalization errorfor each element of the experimental design wasobserved in the range errG ∈ [10−14, 10−6], indi-cating excellent compression accuracy.The coefficients of the N PCEs thus cal-culated are gathered in the compressed designC ={c(1),c(2), ...,c(N)}. After compression, theoriginal representation of each experimental designsample was reduced from n = 105 to a much moremanageable nc = |A d,pc | = 226 scalars.5.4. Compressive PCEClassical PCE is now performed independently foreach of the nc dimensions of C to calculate theaγα coefficients in Eq. (15). For this application,we performed sparse adaptive PCE using the LeastAngle Regression algorithm (LARS) to enforce L1sparsity on the set of coefficients. The polynomialdegree was adaptively chosen for each of the outputdimensions in the range p ∈ [3,15] (for degree-adaptive LARS, see Blatman and Sudret, 2011).The final accuracy of this expansion varied from ac-ceptable to good, with errGγ ∈ [10−4, 10−3].5.5. CPCE resultsFinally, a validation set of Nval = 104 samples ob-tained by crude Monte Carlo simulation is evalu-ated with the full model in Eq. (23) to compare thestatistical properties of the model map with thoseextracted from the coefficients of the CPCE.Expectation value: the reference expectationvalue of the map calculated from the validation setis shown in the left panel of Figure 2. The cor-responding CPCE-based estimate calculated withEq. (16) is shown in the right panel of the same fig-ure. The match between the two is excellent in allthe points in the map coordinates X .Variance: the variance of the map is shown onthe left panel of Figure 3, while the correspond-ing PCE approximation (Eq. (17)) is shown in theright panel of the same Figure. The approximationis once again excellent throughout the domain.Sobol’ indices: the calculation of Sobol’ indicesfor such a large map would be by far too expensivewith classical methods. CPCE, however, providesa functional form of the Sobol’ indices in Eq. (22),which can be used to inexpensively calculate themfor the entire map, even in points that are not in-cluded in the outputs of the original model. Fig-ure 4 shows the full maps of Sobol’ indices on thedomain of X . For validation purposes, we showin Figure 5 the comparison between several point-wise PCEs and CPCE for a slice at x2 ' −0.25.The Sobol’ indices calculated from point-wise PCEare plotted as circles, while the corresponding esti-mates by CPCE by solid lines. The match is excel-lent.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015X 1X 2  −1 −0.5 0 0.5 1−1−0.500.511. 1X 2  −1 −0.5 0 0.5 1−1−0.500.511. 2: Mean map of the model in Eq. (23) as calculated from a reference sampling (left) and its estimate fromPCE coefficients based on Eq. (16) (right).X 1X 2  −1 −0.5 0 0.5 1−1−0.500.512468X 1X 2  −1 −0.5 0 0.5 1−1−0.500.512468Figure 3: Variance map of the model in Eq. (23) as calculated from a reference sampling (left) and its estimatefrom PCE coefficients based on Eq. (17) (right).X 1X 2  −1 −0.5 0 0.5 1−1−0.500.5100. 1X 2  −1 −0.5 0 0.5 1−1−0.500.5100. 1X 2  −1 −0.5 0 0.5 1−1−0.500.5100. 4: First order Sobol’ index maps for ξ1 (left) and ξ2 (center) as estimated with Eq. (22) and correspondingsecond order index map (right).712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015x2  −1 −0.5 0 0.5 S1 PCES1 CPCES2 PCES2 CPCES12 PCES12 CPCEFigure 5: Comparison between Sobol’ indices calcu-lated with point-wise PCE and CPCE on a verticalslice at x2 =−0.25.6. SUMMARY AND CONCLUSIONSIn this paper we have introduced an algorithm thatleverages on the regularity of numerical modelswith a large number of outputs by adopting a com-pression strategy. In particular, it is well known thatPCE has very good sparsity properties for smoothfunctions on bounded domains. Therefore, we in-troduced a two-stage approach to first pre-processthe experimental design of model maps, compress-ing it on a sparse PCE basis, followed by classicalPCE on the compressed experimental design.Due to the linearity of PCE, we could derive sim-ple analytical expressions for the functional repre-sentation of the first moments of the model map interms of its coordinates, as well as for more com-plex (and interesting) quantities that can be nor-mally derived by post-processing PCE coefficients.We want to stress on the fact that this approachdoes not degrade nor improve the convergenceproperties of classical point-wise PCE. This is im-portant, as CPCE does not aim at extending the va-lidity class of PCE methods, but rather at allow-ing the analyst to handle models with a large num-ber of outputs. It is common engineering practicee.g. to calculate expensive FEM responses of com-plex structures, but then to only consider few se-lected quantities of interest for their actual analy-sis (e.g. selected inter-storey displacements, etc.).With CPCE, the entire set of displacements couldbe meta-modelled, hence taking advantage of thecomplexity of the full calculation.Finally, it should be noted that the choice of PCEas the compression tool is just one of many: theonly requirement for the entire formal setting of thispaper is that the spectral representation in Eq. (12)holds. Polynomials are one of a number of or-thonormal bases that can be built to represent a d-dimensional scalar function as a linear superimpo-sition of terms. Other commonly employed spec-tral techniques include Fourier transforms, orthog-onal wavelet decomposition, Karhunen-Loève ex-pansions and many others. Note that standard finiteelement representation is also part of this frame-work. In other words, any spectral decompositionon an orthonormal basis onto which each realiza-tion of the model map is smooth can be used withinthis framework with the same equations.Future extensions of this work include the com-bination of the prediction errors from the two stagesof PCE to the final map, as well as a more efficientcompression strategy that only considers significantterms in the truncated set A d,pc .REFERENCESBlatman, G. and Sudret, B. (2010). “An adaptive al-gorithm to build up sparse polynomial chaos ex-pansions for stochastic finite element analysis.”Prob. Eng. Mech., 25(2), 183–197.Blatman, G. and Sudret, B. (2011). “Adaptivesparse polynomial chaos expansion based onLeast Angle Regression.” J. Comput. Phys, 230,2345–2367.Blatman, G. and Sudret, B. (2013). “Sparse polyno-mial chaos expansions of vector-valued responsequantities.” Proc. 11th Int. Conf. Struct. Saf. Re-liab. (ICOSSAR’2013), New York, USA, G. Deo-datis, ed.Sobol’, I. (2001). “Global sensitivity indices fornonlinear mathematical models and their MonteCarlo estimates.” Math. Comput. Simul., 55(1-3),271–280.Sudret, B. (2007). “Uncertainty propagation andsensitivity analysis in mechanical models - Con-tributions to structural reliability and stochasticspectral methods.” Habilitation thesis, UniversitéBLAISE PASCAL - Clermont II.Sudret, B. (2008). “Global sensitivity analysis us-ing polynomial chaos expansions.” Reliab. Eng.Sys. Saf., 93, 964–979.8


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