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

Data-driven polynomial chaos basis estimation Spiridonakos, Minas D.; Chatzi, Eleni N. 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Data-Driven Polynomial Chaos Basis EstimationMinas D. SpiridonakosPost-doctoral researcher, Inst. of Structural Engineering, D-BAUG, ETH, Zurich,SwitzerlandEleni N. ChatziProfessor, Inst. of Structural Engineering, D-BAUG, ETH, Zurich, SwitzerlandABSTRACT: A non-intrusive uncertainty quantification scheme based on Polynomial Chaos (PC) basisconstructed from available data is introduced. The method uses properly parametrized basis functionsin order to let them adapt to the given input-output data instead of predefining them based on the prob-ability density function of the uncertain input variable. Model parameter estimation is effectively dealtwith through a Separable Non-linear Least Squares (SNLS) procedure that allows the simultaneous es-timation of both the PC basis and the corresponding coefficients of projection. Method’s effectivenessis demonstrated through its application to the uncertainty propagation modelling in two examples: anonlinear differential equation with uncertain initial conditions and a nonlinear single degree-of-freedomsystem with an uncertain parameter. Comparisons with classical PC expansion modelling based on theWiener-Askey scheme are used to illustrate the method’s performance and potential advantages.Polynomial Chaos (PC) expansion has beendemonstrated to effectively model uncertaintypropagation in a number of engineering problems.The main advantage of the PC representation is itslow computational cost, compared to that of tra-ditional approaches such as the classical Monte-Carlo, and its ease of use for model-based analysis,e.g. for the purposes of statistical characterizationof the output, reliability and global sensitivity anal-ysis.Nevertheless, PC expansion for uncertaintyquantification in real applications remains challeng-ing mainly due to problems arising from the sta-tistical characterization of the input variables, aswell as the process of determining a sparse PC ba-sis, in the sense of including only a small num-ber of basis functions which may still providehigh approximation accuracy. With regard to thefirst problem, although Xiu and Karniadakis (2002)have extended the initially proposed PC of Gaus-sian processes on Hermite polynomials proposedby Wiener, to a number of common continuousand discrete Probability Density Functions (PDFs)through the Wiener-Askey scheme, estimating thestatistical distribution of input variables may be anontrivial task since bounded, multi-modal, or dis-continuous PDFs may be found to best fit givenmeasured data (Oladyshkin and Nowak (2012)). Insuch cases, fitting the given data to a common PDFmay significantly reduce the accuracy of the expan-sion, while on the other hand transformations tostandard PDFs normally lead to slower convergencerates.The crucial problem of selecting specific PC sub-spaces is also an open problem that has been treatedin a number of recent studies (for example see Blat-man and Sudret (2011)), with a common approachbeing the forward selection procedure which buildsup the PC model by adding bases till no furtherimprovement is achieved, according to a predeter-mined criterion. In most cases however, such ap-proaches require the estimation of a large numberof candidate models.Moreover, potential limitations of the Wiener-Askey expansion scheme arise in situations wherediscontinuities or complex relationships character-ize the dependency of the output variable on therandom input data (Le Maître et al. (2004)).112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The aim of the present study is to circumvent theaforementioned difficulties associated with PC ba-sis construction. Toward this end, the PC is notbased on basis functions of a fixed form, but in-stead we use orthogonal B-splines functions witha-priori unknown properties that may be adaptedto the specific random input variable characteris-tics. This is accomplished through appropriate ba-sis function parametrization which allows for di-rect estimation of the splines knots, and a Sepa-rable Nonlinear Least Squares (SNLS) type proce-dure (Golub and Pereyra (2003)) that achieves si-multaneous estimation of the basis functions andthe coefficients of projection through a reduced di-mensionality, constrained non-quadratic optimiza-tion problem. The method’s effectiveness is exam-ined via a Monte Carlo study and comparisons withthe classical PC method based on the Wiener-Askeyscheme are made.1. POLYNOMIAL CHAOS EXPANSIONPCE based on the Wiener-Askey scheme concernsthe expansion of a random output variable on poly-nomial chaos basis functions which are orthonor-mal to the probability space of the system’s ran-dom inputs. More specifically, let us consider asystem S which has M random input parame-ters represented by independent random variables{Ξ1, . . . ,ΞM}, gathered in a random vector Ξ ofprescribed joint PDF pΞ(ξ ) (Blatman and Sudret(2011)).The system output, denoted by Y = S (Ξ) willalso be random. Provided that Y has finite variance,it can be expressed as follows:Y =S (Ξ) = ∑j∈Nc jφd( j)(Ξ) (1)where c j are unknown deterministic coefficients ofprojection, d is the multivariate index of the PCbasis, and φd( j)(Ξ) are the PC functions orthonor-mal to pΞ(ξ ). These basis functions φd( j)(Ξ) maybe constructed through tensor products of the cor-responding univariate functions associated throughthe corresponding probability distributions (Xiuand Karniadakis (2002)).As already mentioned, the main drawback of theclassic PC expansion, especially for real-data appli-cations, is the statistical characterization of the in-put variables and the selection problem related withthe choice of an appropriate functional subspace.Even if this is only a small price to pay for ren-dering the estimation of an uncertainty propagationmodel into a deterministic estimation problem, thisselection is of crucial importance for accurate mod-elling. Indeed, despite the fact that theoretically any“extended” (of high dimensionality) PC subspacemay achieve good tracking of the parameter evolu-tion, this is not true when we have to select only asmall number of functions due to reasons of statis-tical efficiency and model parsimony (economy ofrepresentation).On the contrary, the use of PC basis adapted onthe selected input variables data is proposed in thisstudy. Toward this end, the basis functions are de-termined by an a-priori unknown vector of param-eters δ which has also to be estimated along withthe coefficients of projection θ . In this way, thebasis may be automatically adapted on the data inorder to achieve higher accuracy in the case the un-certainty propagation is highly nonlinear, leading toan output which does not follow the distribution ofthe input. In the sequel, a data-driven PC basis es-timation framework based on B-spline functions isdescribed. Based on the attractive properties of theB-splines functions, both smooth or abrupt uncer-tainty propagation relationships may be efficientlyexpanded on the constructed basis.1.1. Adaptable B-spline functionsThe values of the input variable Ξ are consideredto be samples drawn from a PDF which may beapproximated by a continuous piecewise polyno-mial function of order k (de Boor, 2001, Chs. 7-8). Then, according to the theorem of Curry andSchoenberg (de Boor, 2001, pp. 97-98), a basis ofsplines (B-splines) may be constructed for the cor-responding piecewise polynomial space. By con-sidering, for the purposes of simplicity, a boundedPDF and the univariate case, the B-splines are fullydefined by an appropriate nondecreasing sequenceof points (knots) τ = [τ1, . . . ,τp+k] ∈ [α,β ] ⊂ ℜ(where p is the basis dimensionality). The afore-mentioned theorem leaves open the selection of thefirst k and last k knots. However, imposing no conti-212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015nuity conditions at the endpoints, τ1 = . . .= τk = αand τp+1 = . . . = τp+k = β may be selected. Thischoice is also consistent with the fact that the con-structed basis provides a valid representation onlyfor the interval [τk,τp+1], that is [α,β ].Thus, in terms of the above quantities theB-splines of order k may be described bythe functional subspace parameter vector δ ∆=[τk+1, . . . ,τp]T of dimension dim(δ ) = p− k con-sisting of the non-decreasing sequence of the freeinternal knots. The variable Y =S (Ξ) is then ex-pressed as:Y =p∑j=1c j ·φ kj (Ξ,δ ) (2)where φ kj (Ξ,δ ) denotes the sequence of B-splinesof order k. There are several ways to define theB-spline functions φ kj (ξ ,δ ) for a given realizationξ . A convenient one is by the means of the Cox-de Boor recursion formula for the normalized B-splines (de Boor, 2001, p. 90):φ1j (ξ ,δ ) ={1 if τ j ≤ ξ < τ j+10 otherwise(3a)φ ij(ξ ,δ ) = w j,i(ξ ) ·φ i−1j (ξ ,δ )++(1−w j,i(ξ )) ·φ i−1j+1(ξ ,δ ), for 1 < i≤ k(3b)wherew j,i(ξ ) ={ ξ−τ jτ j+i−1−τ j if τ j < τ j+i−1,0 otherwise.(3c)B-splines are characterized by a number of prop-erties which make them particularly attractive fordata-driven PC expansion applications. A signif-icant property of B-splines is their local support,that is φ kj [ξ ,δ ] 6= 0 only for ξ ∈ [τ j,τ j+k). Due tothis local support, the resulting basis may consistof splines with various characteristics and thereforemay be capable of tracking parameters with mixedtype of evolution, that is alternating patterns ofsmooth and abrupt changes of the PDF. Moreover,B-splines are locally linearly independent, that isthey provide a basis for the piecewise polynomialspace even for an interval [α,β ] ⊆ [τ1,τp+k]. Fi-nally, by selecting the proper B-splines order k, var-ious degrees of smoothness may be achieved. Forexample, for k = 1 the basis consists of piecewiseconstant functions, for k = 2 linear B-splines, fork = 3 quadratic, for k = 4 cubic, and so on. Yet,smoothness may also be controlled by the proxim-ity of knots (de Boor, 2001, Ch. 9). A thoroughanalysis of B-splines and their properties may befound in de Boor (2001).Constraints on δ . The internal knots τk+1, . . . ,τpform a nondecreasing sequence of real numbers de-fined in the open set (α,β ). Thus, they have to sat-isfy proper constraints related with their bounds andtheir order relation.Simple constraints may be defined for the case ofinternal knots with multiplicity one. This is not re-strictive in view of the fact that a B-spline with aninternal knot of multiplicity m > 1 may be approx-imated by replacing this knot by m simple knotsnearby (de Boor, 2001, p. 106).Thus, an appropriate order relation constraint,suitable for numerical purposes (as it prevents theknots from coalescing when their distance becomespractically equal to zero) is:τ j− τ j−1 > ε, j = k+1, . . . , p+1, (4)where 0 < ε  (β −α) is a selected, sufficientlysmall separation parameter.Therefore, the following p−k+1 inequality con-straints are imposed on the parameter vector δ :τk+1− τk = τk+1−α ≥ ετk+2− τk+1 ≥ ε...τp− τp−1 ≥ ετp+1− τp = β − τp ≥ ε=⇒−1 0 . . . 0 01 −1 . . . 0 0....... . .......0 0 . . . 1 −10 0 . . . 0 1 ·τk+1τk+2...τp−1τp︸ ︷︷ ︸δ−−α− ε−ε...−εβ − ε≤ 0(5)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015which are also depicted schematically in Fig. 1.α                                                           β=...==...=Figure 1: Constraints imposed on δ . The distance be-tween two internal knots has to be larger than ε . Thefirst and last internal knots have also to be ε away fromthe endpoints α and β , respectively.1.2. PCE model parameter estimationThe proposed data-driven PCE model estimationproblem involves the estimation of the parame-ter vector δ consisting of the B-splines internalknots and the coefficients of projection vector cfrom the available N-samples long set of data ξ =[ξ1, . . . ,ξN ]T and y = [y1, . . . ,yN ]T . Using the B-spline functions of Eq. (3) (for given k and p), thePCE model may be re-written in the following non-linear regression form:yn =p∑j=1c j ·φ kj (ξn,δ )+ en(δ ,c)⇒⇒ yn =φ k1 (ξn,δ )φ k2 (ξn,δ )...φ kp(ξn,δ )︸ ︷︷ ︸φ (ξn,δ )(p×1)T·c1c2...cp︸ ︷︷ ︸c(p×1)+en(δ ,c)(6)or equivalently by using the stacked signal and in-novations sequence vectors y =[y1, . . .yN]Tande(δ ,c) =[e1(δ ,c), . . . ,eN(δ ,c)]T:y =Φ(ξ ,δ ) · c+ e(δ ,c) (7)where Φ(ξ ,δ ) =[φ k1(ξ1,δ ), . . . ,φ kp(ξN ,δ )]T.The estimation of the model parameter vectorsδ and c may be based on the minimization of thePrediction Error (PE) criterion V (δ ,c) = ‖e(δ ,c)‖2consisting of the sum of squares of the model’s er-rors, subject to the constraints discussed in the pre-vious section which guarantee the linear indepen-dence of the basis functions, that is:[δˆT , cˆT ]T = argminδ ,cV (δ ,c) = argminδ ,c‖e(δ ,c)‖2subject to H(δ )≤ 0(8)In this relation, argmin stand for “argument mini-mizing”, ‖ · ‖ indicates Euclidean norm, and e(δ ,c)the model error being provided by the model ex-pression of Eq. (6). A hat designates estima-tor/estimate of the indicated quantity.The nonlinear dependence of the basis functionsφ kj (ξ ,δ ) on the parameter vector δ renders the PE-based estimation problem a nonlinear optimizationproblem with linear inequality constraints. Thisproblem may be solved through common iterativeconstrained nonlinear optimization methods withrespect to the dim(c)+dim(δ ) = p+ p− k param-eters and by utilizing the gradient of the objectivefunction V (δ ,c).However, by taking into consideration the factthat the parameters c and δ form two completelydisjoint sets more efficient estimation methods maybe derived. Such problems are referred to asSeparable Nonlinear Least Squares (SNLS) prob-lems and an efficient method for their solution isthe Variable Projection (VP) method introducedin early seventies by Golub and Pereyra (1973),which is based on the fact that c appears linearlyin the model function Φ(ξ ,δ ) · c. Thus, if we as-sume known nonlinear parameters δ , the cˆ estimatemay be readily obtained through the ordinary leastsquares equation:cˆ =(ΦT (ξ ,δ ) ·Φ(ξ ,δ ))−1ΦT (ξ ,δ ) · y⇒⇒ cˆ =Φ†(ξ ,δ ) · y (9)with † denoting pseudo-inverse. Hence, the modelerrors may be obtained ase(δ ,c) = y−Φ(ξ ,δ ) · c ⇒⇒ eVP(δ ) = y−Φ(ξ ,δ ) ·Φ†(ξ ,δ ) · y⇒⇒ eVP(δ ) =(I−Φ(ξ ,δ ) ·Φ†(ξ ,δ ))· y (10)412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015and as a consequence the optimization problem ofEq. (8) takes the variable projection functionalform:δˆ = argminδ‖(I−Φ(ξ ,δ ) ·Φ†(ξ ,δ ))· y‖2subject to H(δ )≤ 0(11)with H(δ ) denoting the linear constraint functional.In this way, the nonlinear parameters δ may be esti-mated through Eq. (11) by the means of constrainednonlinear optimization techniques with only p− kunknown parameters, while c may be subsequentlyestimated through linear least square optimization(Eq. (9)). The method is referred to as VP methodas the matrix(I −Φ(ξ ,δ ) ·Φ†(ξ ,δ ))is the pro-jector on the orthogonal complement of the columnspace of Φ(ξ ,δ ) (Golub and Pereyra (2003)).The minimization of the VP functional entailsa number of advantages compared to the originalminimization problem. The most important is thedimensionality reduction of the nonlinear optimiza-tion problem – while this reduction does not implychanges to the stationary points (minima and max-ima) of the original problem. This holds under therather mild condition of constant (not necessarilymaximum) rank of the regression matrix Φ(ξ ,δ )over the whole parameter search space of δ . How-ever, in our case the constraints described in the pre-vious section define an appropriate parameter spacein which the constant full rank of Φ(ξ ,δ ) is guar-anteed.Nonetheless, the cost for reducing the dimen-sion of the nonlinear optimization problem is theincreased complexity of the VP objective functionVVP(δ ) gradient computation. Golub and Pereyrain their aforementioned study derived the necessaryrelationships for the differentiation of a pseudoin-verse matrix and concluded to the required gradi-ent with respect to the nonlinear parameter vector.Approximate solutions which aim at computationaltime reduction have also been proposed in the liter-ature. A comparison and asymptotic analysis studyfor the three most frequently used algorithms maybe found in Ruhe and Wedin (1980). In the presentstudy we follow the Golub and Pereyra exact ap-proach.2. NUMERICAL CASE STUDY2.1. Test case IFor this first example we consider the following dif-ferential equation governing the movement of a par-ticle being under the influence of a potential fieldand a friction force (Le Maître et al. (2004)):d2Xdt2+2dXdt=−352X3 +152X (12a)with uncertain initial initial positionX(t = 0,ξ )= 0.05+0.2ξ , ξ ∼U (−1,1) (12b)and vanishing velocitydXdt∣∣∣∣t=0= 0 (12c)The analytical prediction of the steady state po-sition of the particle is given by{X(t→ ∞,ξ ) =−√15/35, ξ <−0.25,X(t→ ∞,ξ ) =√15/35, ξ >−0.25,(13)This uncertainty propagation problem was solvedin Le Maître et al. (2004) through the Galerkin ap-proach and the proposed therein Wiener-Haar ex-pansion model. Nonetheless, in the present studythe non-intrusive regression problem of the initialconditions uncertainty propagation to the steadystate solution (for t = 10 s) is considered, that is Y =X(t = 10s,ξ ). For this reason Eq. (12) is simulatedfor N = 100 times, with the corresponding sam-ples of the uncertain parameter ξ being drawn fromthe standard uniform distribution through the LatinHypercube Sampling (LHS) method (Helton andDavis (2003)). A Runge-Kutta scheme is utilizedfor performing time integration with a time step∆t = 0.001 s. The output yn(n = 1, . . . ,100) ver-sus the corresponding input variables ξn are shownin Fig. 2.The criterion of the normalized sum of squarederrors:NSE =∑Nn=1 e2n∑Nn=1 y2n(%) (14)is utilized for evaluating the performance of the dif-ferent expansion schemes. The calculated NSE val-ues of the data-driven expansion based on adapt-able B-spline basis of various orders k = 1, . . . ,4 is512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015−1 −0.5 0 0.5 1−0.8−0.6−0.4−ξYFigure 2: Steady-state position of the moving particleY = X(t = 10s,ξ ) versus the corresponding inputvariable ξ for the 100 simulations conducted.shown in Fig. 3 along with those obtained by theclassical PC expansion based on Legendre polyno-mials. For all cases, the maximum number of baseshas been limited to 10. It is noted that a constraintparameter ε equal to 0.02 is selected for the data-driven approach while the SNLS optimization isinitialized by values estimated through a gradient-free Particle Swarm Optimization (PSO) algorithm(Engelbrecht (2006)) . This extra step is taken in or-der to reduce the possibility of wrong convergenceof the optimization procedure to local minima dueto arbitrary initialization.The results of Fig. 3 reveal superior performanceof the proposed method since the discontinuity ofthe input-output relationship is correctly captured,as expected, by the B-splines with k = 1 and onlytwo bases, that is a single internal knot estimatedto be at ξ = −0.2678. On the other hand, a largenumber of Legendre polynomials are necessary toachieve similar accuracy.2.2. Test case IIIn this second example the test case of a hystereticdissipative SDOF system is examined. The SDOFsystem is actually a mass-damper systems with anadditional element producing a nonlinear restoringforce F(t) described by the Bouc-Wen model (Fig.4):mdX2dt+ cdXdt+F(t) =U(t) (15a)F(t) = λk`X(t)+(1−λ )k`Z(t) (15b)1 2 3 4 5 6 7 8 9 1010−810−610−410−2100102Number of basesNSE (%)  B−splines (k = 1) B−splines (k = 2) B−splines (k = 3) B−splines (k = 4) LegendreFigure 3: Test case I: NSE criterion values for the ex-pansion models based on classical PC (Legendre poly-nomials) and the introduced data-driven method (B-splines of order k = 1, . . . ,4).uxFigure 4: Single degree of freedom system with hys-teretic restoring force.withZ˙(t) = AX˙(t)+β∣∣∣∣dXdt∣∣∣∣ |Z(t)|n−1Z(t)− γ dXdt|Z(t)|n(15c)λ designating the post- to pre-yield stiffness k` ra-tio, and A, β > 0, γ and n the dimensionless quanti-ties controlling the shape of the hysteresis loop (Is-mail et al. (2009)).The nonlinear SDOF system is considered to besubject to a static random force equal to 100 N,while it is characterized by uncertainty of the pa-rameter λ ≡ ξ ∼ U (0,1). The properties of thesystem are summarized in Table 1.Again the regression problem of the uncertaintypropagation to the steady-state displacement (fort = 20 s) of the SDOF hysteretic system is con-sidered, that is Y = X(t = 20s,ξ ). The system issimulated N = 100 times, with the equations of mo-tion describing the system being integrated through612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 1: Properties of the SDOF system with hystereticrestoring force.mass m = 1 kgdamping coefficient c = 1 N/ (m/s)linear stiffness coefficient k` = 100 N/mpost- to preyield stiffness ratio λ ∼U (0,1)hysteretic loop shape parameters A = 1,n = 3β = 1,γ = 10 0.2 0.4 0.6 0.8 1050100150200250300350400ξYFigure 5: Steady-state displacement of the hystereticSDOF system Y = X(t = 20s,ξ ) versus the correspond-ing input variable ξ for the 100 simulations conducted.a Runge-Kutta scheme with a time step ∆t = 0.01s. The values of the output variable versus those ofthe uncertain input variable are shown in Fig. 5.The normalized sum of squared errors criterionachieved by the classical PCE based on Legendrepolynomials for various degrees p = 1, . . . ,10 iscontrasted to those of a data-driven expansion basedon B-splines of various orders k = 1, . . . ,4 in Fig. 6.As clearly shown the data-driven expansion outper-forms the classical PC model for B-splines order ofk = 3 and k = 4, yet with a much larger convergencerate.3. CONCLUSIONSA data-driven uncertainty quantification schemehas been introduced. The method is based onproper basis function parametrizations and anSNLS type procedure which allows the efficient so-lution of the parameter estimation problem. Themethod’s effectiveness has been assessed throughtwo uncertainty propagation representation testcase studies. Comparisons with classical PCEbased on the Wiener-Askey scheme, demonstrated1 2 3 4 5 6 7 8 9 1010−210−1100101102Number of basesNSE (%)  B−splines (k = 1) B−splines (k = 2) B−splines (k = 3) B−splines (k = 4) LegendreFigure 6: Test case II: NSE criterion values for theexpansion models based on classical PC (Legendrepolynomials) and the introduced data-driven method(B-splines of order k = 1, . . . ,4).the method’s advanced capabilities and superiorperformance characteristics. Future work will in-clude the comparison of the method with other datadriven methods such as the arbitrary PC (aPC) in-troduced in Oladyshkin and Nowak (2012), and theextension of the method for the intrusive PC ap-proach.4. REFERENCESBlatman, G. and Sudret, B. (2011). “Adaptive sparsepolynomial chaos expansion based on least angle re-gression.” Journal of Computational Physics, 230(6),2345– Boor, C. (2001). A Practical Guide to Splines.Springer-Verlag, Revised edition.Engelbrecht, A. P. (2006). Fundamentals of Computa-tional Swarm Intelligence. Wiley.Golub, G. and Pereyra, V. (2003). “Separable nonlinearleast squares: The variable projection method and itsapplications.” Inverse Problems, 19, R1–R26.Golub, G. H. and Pereyra, V. (1973). “The differentationof pseudo-inverses and nonlinear least squares prob-lems whose variables separate.” SIAM Journal on Nu-merical Analysis, 10(2), 413–432.Helton, J. C. and Davis, F. J. (2003). “Latin hypercubesampling and the propagation of uncertainty in analy-ses of complex systems.” Reliability Engineering andSystem Safety, 81(1), 23–69.Ismail, M., Ikhouane, F., and Rodellar, J. (2009). “Thehysteresis Bouc-Wen model, a survey.” Archives of712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Computational Methods in Engineering, 16(2), 161–188.Le Maître, O. P., Knio, O. M., Najm, H. N., and Ghanem,R. G. (2004). “Uncertainty propagation using Wiener-Haar expansions.” Journal of Computational Physics,197(1), 28–57.Oladyshkin, S. and Nowak, W. (2012). “Data-driven un-certainty quantification using the arbitrary polynomialchaos expansion.” Reliability Engineering & SystemSafety, 106, 179–190.Ruhe, A. and Wedin, P. Å. (1980). “Algorithms for sep-arable nonlinear least squares problems.” SIAM Re-view, 22, 318–337.Xiu, D. and Karniadakis, G. E. (2002). “The Wiener-Askey polynomial chaos for stochastic differentialequations.” SIAM Journal on Scientific Computing,24(2), 619–644.8


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