12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Polynomial Chaos Expansions For Damped OscillatorsChu V. MaiPhD student, ETH Zürich, Institute of Structural Engineering, Chair of Risk, Safety &Uncertainty Quantification, Stefano-Franscini-Platz 5, CH-8093 Zürich, SwitzerlandBruno SudretProfessor, ETH Zürich, Institute of Structural Engineering, Chair of Risk, Safety &Uncertainty Quantification, Stefano-Franscini-Platz 5, CH-8093 Zürich, SwitzerlandABSTRACT: Uncertainty quantification is the state-of-the-art framework dealing with uncertainties aris-ing in all kind of real-life problems. One of the framework’s functions is to propagate uncertainties fromthe stochastic input factors to the output quantities of interest, hence the name uncertainty propagation.To this end, polynomial chaos expansions (PCE) have been effectively used in a wide variety of practicalproblems. However, great challenges are hindering the use of PCE for time-dependent problems. Moreprecisely, the accuracy of PCE tends to decrease in time. In this paper, we develop an approach basedon a stochastic time-transform, which allows one to apply low-order PCE to complex time-dependentproblems.1. INTRODUCTIONUncertainty quantification has become a key topicin modern engineering in the last decade due to theincreasing complexity of physical systems and as-sociated computational models. One of the objec-tives of the framework is to propagate uncertain-ties from the stochastic input factors of the modelto the output quantities of interest, hence the nameuncertainty propagation. For this purpose, poly-nomial chaos expansions (PCE) have been widelyused as approximate models (or metamodels) thatsubstitute computationally expensive ones.PCE, however, face challenges when applied totime-dependent problems, e.g. involving structuralor fluid dynamics or chemical systems. The great-est challenge hindering the use of PCE is the de-crease of the accuracy in time (Ghosh and Iac-carino, 2007; Le Maître et al., 2010).The causes of the decaying accuracy of PCE intime-dependent problems can be classified into anapproach-related cause and an inherent cause. Theapproach-related cause refers uniquely to the in-trusive approach, which requires manipulation ofthe mathematical equations describing the consid-ered problem. In particular, the intrusive approachsolves a system of reformulated differential equa-tions, which are derived from the original systemby substituting PCE for the quantity of interest (LeMaître et al., 2010). The error due to the approx-imate solution is accumulated over time and willcertainly be excessive at some point. In contrast,the non-intrusive approach considers the compu-tational model as a black-box model in which re-sponses at different instants can be examined in-dependently, which prevents the accumulation oferror at late instants. The inherent cause refers tothe increasing complexity in the system’s responsewith time. Often, as time elapses, the relationshipbetween the output quantity and the input factorsbecomes increasingly complicated, i.e. non-linear,non-smooth or discontinuous. This makes the rep-resentation of the system’s response with PCE in-creasingly hard.Existing approaches in literature for dealing withtime-dependent problems may be classified in dif-ferent categories. The high-order PCE approachincludes the decomposition of the complex prob-lem into simpler sub-problems (non-intrusive low-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015rank separated approximations (Doostan et al.,2013), intrusive reduced PCE (Pascual and Ad-hikari, 2012), factorized (resp. logarithmic) poly-nomial dimensional decomposition F-PDD (resp.L-PDD) (Yadav and Rahman, 2013)). It also in-cludes methods for reducing the size of the high-order PCE basis (sparse PCE (Blatman and Sudret,2011), two-step PCE (Peng et al., 2010)) and ad-vanced computational techniques for computing thehigh-order PCE (e.g. decoupled PCE (Pham et al.,2014)).The local PCE approach consists in dividing theinput space into subspaces according to the detecteddiscontinuities or dissimilarities. One then buildsa local PCE in each subspace and combines thoselocal PCE models to obtain a global metamodel.This approach includes multi-element PCE (Wanand Karniadakis, 2006b), simplex stochastic collo-cation method (Witteveen and Iaccarino, 2013) andmixture of PCE (Nouy, 2010).There exist additional approaches involving themodification or update of the polynomial chaos ba-sis: enriched PCE (Ghosh and Ghanem, 2008),time-dependent PCE (Gerritsma et al., 2010), flowmap composition (Luchtenburg et al., 2014), hybridPCE (Heuveline and Schick, 2014), dynamicallybi-orthogonal decomposition (Choi et al., 2014)and wavelet-based Wiener-Haar expansion (Sahaiand Pasini, 2013).An attractive approach that was proposed re-cently is to transform the response trajectories inorder to make the relationship between the outputand the input factors less complex. Witteveen andBijl (2008) and Desai et al. (2013) represented thedynamic response trajectories as functions of thephase φ instead of time t in order to obtain in-phasevibrations. The phases are extracted from the ob-servations based on the local extrema of the time se-ries. The response trajectories are then transformedfrom time-histories to phase-histories. Finally PCEare applied in the phase space. This approach relieson the assumption that the phase is well defined asa function of time t. In case this assumption doesnot hold, the general solution will not be straight-forward.Le Maître et al. (2010) represented the responsesin a rescaled time τ such that the dynamic responsesvary in a small neighborhood of a reference trajec-tory. The reference trajectory is chosen so that thevariability of the uncertain parameters does not in-duce significant changes in the trajectory. In gen-eral, the scaled time τ is not a linear function of t,i.e. the "clock speed" τ˙ = dτdt is not constant. Thisrelation depends on the difference between the dy-namic response and the reference trajectory. In thementioned paper, τ˙ is adjusted in an intrusive wayat each step so that the difference is minimized.As a summary, PCE fail to represent long-termtime-dependent system responses because of theirinherent increasing complexity. So far, there hasnot been a versatile tool that helps overcome theproblem. The objective of this paper is to intro-duce a non-intrusive approach that allows the use ofPCE for time-dependent problems. The proposedapproach relies on a non-intrusive stochastic time-transform of the response trajectories which aims atmaximizing the similarities in frequency and phasecontent of sampled time-histories.The paper is organised as follows: in Section 2,the fundamentals of PCE are recalled. In Section 3,we propose a non-intrusive PCE approach for time-dependent problems and use a benchmark exampleof stochastic dynamics to illustrate it.2. POLYNOMIAL CHAOS EXPANSIONSLet us consider a computational model Y =M (X )where X = (X1, . . . ,XM)T is a M-dimensional ran-dom input vector with the probability density func-tion fX defined in the probability space (Ω ,F ,P).Without loss of generality, we investigate the casewhen the input random variables are independent.Assume that the scalar output Y is a second orderrandom variable, i.e. E[Y 2]< +∞. Then, Y be-longs to a Hilbert space H of square-integrablefunctions of random vector X . Denote by Hi theHilbert space associated with the marginal proba-bility measure PXi(dxi) = fXi(xi)dxi, in which oneselects a basis of associated orthonormal univari-ate polynomial functions {ψ ik,k ∈ N}. For in-stance, when Xi is a uniform (resp. standard nor-mal) random variable, those are orthonormal Leg-endre (resp. Hermite) polynomials. The general-ized polynomial chaos representation of Y can be212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015written as follows (Xiu and Karniadakis, 2002):Y = ∑α∈NMyαψα (X ) (1)in which ψα (X ) = ψ1α1(X1) . . .ψMαM(XM) are mul-tivariate orthonornal polynomials with respect tofX (x), α = (α1, . . . ,αM) are multi-indices with{αi, i = 1, . . . ,M} denoting the degree of the uni-variate polynomial of Xi and yα ’s are the corre-sponding polynomial chaos coefficients or "coordi-nates" of Y in the space spanned by the polynomialchaos basis.In practice, the use of infinite series as in Eq. (1)is not tractable and thus, an approximate truncatedrepresentation is utilized:Y ≈ ∑α∈Ayαψα (X ) (2)in which A is the set of truncated multi-indices α .Different choices can be made for the truncationscheme. For instance, the total degree of the expan-sion is set to be not larger than a prescribed value,i.e. :A = {α ∈NM : ‖α ‖1=α1+ . . .+αM 6 p} (3)The computation of the coefficients yα can beperformed by means of the intrusive approach (i.e.spectral stochastic finite element method (Ghanemand Spanos, 2003)) or non-intrusive approaches(e.g. projection, regression methods). The readeris refered to Sudret (2007) for further details. In thefollowing, we will use the adaptive sparse polyno-mial chaos expansions (Blatman and Sudret, 2011),which is a non-intrusive approach based on least-square minimization. The accuracy of PCE is es-timated by means of the leave-one-out error (Blat-man, 2009).3. POLYNOMIAL CHAOS EXPANSIONSFOR TIME-DEPENDENT PROBLEMSAn intrusive time-transform of the trajectories wasproposed by Le Maître et al. (2010) aiming at rep-resenting the time-histories in a small neighbor-hood of a reference trajectory so as to reduce theirvariability. This is done by minimizing the Eu-clidean distance between the distinct trajectoriesand a reference counterpart. Herein, we proposea non-intrusive transform, which consists in find-ing a suitable stochastic mapping of the time linethat increases the similarity between different tra-jectories, thus allowing low-order PCE to be ap-plied. "A neighborhood of a reference trajectory"(Le Maître et al., 2010) is characterized by similar-ity in terms of distance, whereas "in-phase vibra-tions" (Witteveen and Bijl, 2008) are characterizedby similarity in terms of frequency and phase con-tent. In the current paper, the proposed approach fo-cuses on increasing the similarity both in frequencyand phase.3.1. Proposed approachConsider a time-dependent system (e.g. a struc-tural dynamic or chemical system) whose responseis modelled by a system of first-order ordinary dif-ferential equations:dxdt= f (x,ξ , t) (4)where t ∈ [0,T ], the initial conditions x(t = 0) =x0 are deterministic and the random vector ξ de-notes the uncertain parameters governing the sys-tem behavior, e.g. masses, stiffness, reaction pa-rameters. ξ comprises independent second-orderrandom variables defined in the probability space(Ω ,F ,P). Note that when a non-intrusive ap-proach is used, it is not required to know explic-itly the equations, i.e. only runs of the availablesolver for realizations of ξ will be used. The ini-tial conditions can also be uncertain, in which casethey become random variables belonging to ξ . Thetime-dependent response of the system is denotedx(t,ξ ). Without loss of generality, one can con-sider one component of the response, e.g. x(t,ξ )with the initial condition x(t = 0) = x0. At eachtime instant, x(t,ξ ) is a second-order random vari-able. As in Witteveen and Bijl (2008), we assumethat x(t,ξ ) is an oscillatory response.We conduct the time transform operation thatmakes the trajectories similar in frequency andphase content as follows:• One first chooses a reference trajectory xr(t)which is for instance obtained by consider-ing the mean values of the input vector ξ , i.e.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015xr(t) = x(t,ξ r = E [ξ ]). In general, xr(t) maybe any realization of the response quantity x(t)obtained for a specific sample of ξ . For the ex-ample considered in this paper, the choice ofxr(t) shall not affect the final results.• One determines the fundamental frequency frand period Tr of the reference trajectory xr(t).We emphasize that herein xr(t) is assumed tobe a monochromatic signal, i.e. fr is the onlyfrequency contained in xr(t). The case whenx(t,ξ ) is multichromatic will be consideredin a future work. To estimate fr and Tr, onecan use two techniques. The first technique isbased on the Fourier transform of xr(t), i.e. itis applied in the frequency domain. The fre-quency corresponding to the peak of the fre-quency spectrum of xr(t) is the expected valueof fr and Tr = 1/ fr. The second techniqueis based on directly examining the trajectoryof xr(t) in the time-domain. One measuresthe average time interval between consecutivemaxima (or minima) of xr(t) which is the ap-proximate value of Tr. This quick and simpleestimation is found sufficiently effective in theexample in this paper.• The following pre-processing consists in trans-forming the time line with the purpose of in-creasing the similarity between different real-izations of the output x(t,ξ ). Assume that oneis given a set of trajectories xi(t)≡ x(t,ξ i), i =1, . . . ,n for n realizations ξ i of the input ran-dom vector. For each i the following steps areperformed:– Determine the fundamental frequency fiand period Ti of the considered trajectoryxi(t).– Define a linear time transform operatorτ = ki t + φi. This operator consists oftwo actions, namely scaling and shifting,respectively driven by parameters ki andφi. The time line is stretched (resp. com-pressed) when ki > 1 (resp. 0 < ki < 1)and is shifted to the left (resp. to theright) when φi < 0 (resp. φi > 0). In gen-eral, the time transform is defined basedon the considered problem and a non-linear transform shall be possible. Whenx(t,ξ ) is monochromatic, a linear trans-form is however sufficient.– Determine (ki, φi) as the global maxi-mizer of the similarity measure betweenthe considered trajectory and the refer-ence counterpart. We propose the follow-ing similarity measure as a function of kand φ :g(k,φ) =∣∣∣∣T∫0xi(kt +φ)xr(t)dt∣∣∣∣‖xi(kt +φ)‖‖xr(t)‖, (5)in whichT∫0xi(kt + φ)xr(t)dt is the in-ner product of the two considered time-histories and ‖ · ‖ is the associated l2-norm, e.g. ‖xr(t)‖ =√T∫0x2r (t)dt. Inpractice, trajectories are discretized overa grid and the inner product (resp. thel2-norm) is reduced to the classical dotproduct of the two considered vectors(resp. the Euclidean length). By Cauchy-Schwarz inequality, this similarity mea-sure always takes values in the interval[0,1]. In fact, if (k,φ) is a solutionthen so is (k,φ ± Tr/2), i.e. when thetransformed-function is shifted by a dis-tance equal to Tr/2 (whether to the leftor to the right), the similarity measurereaches another global maximum (seeFigure 1 for example).To ensure the uniqueness of the solution,constraints on the support of the param-eters need to be imposed. If Tr/4 ≤ φ ≤Tr/2 (resp. −Tr/2 ≤ φ ≤ −Tr/4) com-prises a solution, then φ − Tr/2 (resp.φ +Tr/2) in the interval [−Tr/4,Tr/4] isalso a maximizer. Thus, it is sufficient toconsider φ in the interval [−Tr/4,Tr/4].In addition, the scaling factor k is as-412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015sumed to be positive. Finally, one has:(ki,φi) = arg maxk∈R+|φ |6Tr/4g(k,φ) (6)– Represent xi(t) on the transformed timeline τ . For this purpose, one choosesa grid line of τ with the desired timeinterval. In fact, the finer the grid is,the smaller is the error introduced by theinterpolation. Herein, the time step inthe time line τ is chosen equal to thediscretization step in the original timet. The trajectory xi(t) is projected ontoτi = ki t +φi to obtain xi(τi). Finally, thelatter is linearly interpolated on τ yield-ing xi(τ).• One builds PCE of k(ξ ), φ(ξ ) and x(τ,ξ )using the realizations {ki,φi,xi(τ,ξ ), i =1, . . . ,n} as experimental design (or trainingset). Because k(ξ ) and φ(ξ ) are scalar quan-tities, the computation of their PCE modelsis straightforward. Note that the computationof PCE of the vector-valued response x(τ,ξ )might be computationally expensive when thetime horizon T is large. This computationalcost can be reduced significantly by meansof principal component analysis (Blatman andSudret, 2013). Although we are aware of thisuseful tool, we will not use it in this paper be-cause the considered duration T is relativelyshort (30 s); in addition, the response in thetransformed time line becomes a smooth func-tion of the input variables ξ and requires onlylow-order PCE which are not expensive tocompute.Given a new realization of input parameters ξ 0,one can predict the corresponding responses of thesystem using PCE as follows:• One predicts x(τ,ξ 0), k(ξ 0) and φ(ξ 0) usingthe computed PCE models.• One maps x(τ,ξ 0) into x(t,ξ 0) using t =τ−φ(ξ 0)k(ξ 0)0 510−10100.51kφObjective functionFigure 1: Similarity measure g(k,φ) (see Eq. (5))3.2. ExampleLet us consider a non-linear single-degree-of-freedom Duffing oscillator under free vibration.The equation of motion reads:x¨(t)+2ω ζ x˙(t)+ω2 (x(t)+ ε x3(t)) = 0 (7)with the initial conditions x(t = 0) = 1 and x˙(t =0) = 0. The motion of the oscillator is driven bythree uncertain parameters described as follows:ζ = 0.05(1+0.05ξ1), ξ1 ∼U (−1,1)ω = 2pi(1+0.2ξ2), ξ2 ∼U (−1,1)ε =−0.5(1+0.5ξ3), ξ3 ∼U (−1,1)(8)We aim at building PCE of x(t) as a function of therandom variables ξ = (ξ1,ξ2,ξ3)T.Figure 2 shows 20 trajectories of x(t) from anexperimental design comprising 200 realizations.The trajectories are oscillatory signals, however,they are not periodic. Tenth-order PCE applied di-rectly to this experimental design (i.e. without pre-processing the trajectories) result in the leave-one-out (LOO) error depicted in Figure 3. Although it isacceptable in the early time instants, this error startsto be excessive at t = 5 s. For illustration, we com-pare the prediction of this direct PCE model for aparticular realization ξ 0 = (−0.95,−0.64,−0.89)Twith the corresponding actual response (Figure 4).It is clear that the prediction fails after 5 seconds.We now apply the time-transform approach tothe above problem. Using tenth-order PCE, we canmodel the parameters k and φ accurately. For a val-idation set X ={ξ (1), . . . ,ξ (n)}of size n = 200,Figure 5 depicts the values of k and φ predictedby the PCE versus the actual values. The LOO er-rors of the PCE models of k and φ are 9.3× 10−6512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 20 25 30−1−0.500.51t (s)x(t)Figure 2: Twenty different trajectories x(t) in the origi-nal time line t0 5 10 15 20 25 3010−1010−810−610−410−2100t (s)LOO errorsFigure 3: Evolution of the leave-one-out error of PCEin the original time line t0 5 10 15 20 25 30−1−0.500.51t (s)x Actual modelPCE predictionFigure 4: Prediction of x(t) by PCE vs. actual responsetrajectoryand 5.2× 10−3, respectively. The transformed tra-jectories become in-phase vibrations, as shown inFigure 6. For the sake of comparison, PCE of or-der up to ten are used to model x(τ). However,the analyses show that the optimal order of thePCE to obtain the largest accuracy is around six.The resulting LOO error normalized by the ampli-tude envelope |xe(τ)| is depicted in Figure 7. Onenotices that the LOO error has a similar oscilla-tory feature as observed in Wan and Karniadakis(2006a). The error oscillates around 10−4 and at-tains its minima (resp. maxima) at the instantswhen x(τ) reaches its maximal or minimal ampli-tude (resp. zero amplitude) during an oscillationcycle. For the same validation set X , Figure 8(resp. Figure 9) compares the response trajectoriesin the transformed time line τ (resp. in the orig-inal time line t) predicted by PCE with the actualresponse trajectories for two realizations of the in-put vector, namely ξ 1 = (−0.02,0.47,−0.38)T andξ 1 = (−0.95,−0.64,−0.89)T. The results illus-trate the accuracy of the proposed time-transformapproach.0.8 1 1.2 1.40.811.21.4Actual valuesPCE k−0.1 −0.05 0 0.05 0.1−0.1−0.0500.050.1Actual valuesPCE φFigure 5: Prediction of k and φ by PCE vs. actual val-ues0 5 10 15 20 25 30−1−0.500.51τ (s)x(τ)Figure 6: Twenty different trajectories x(τ) in the trans-formed time line τ . The red curves are the amplitudeenvelopes of the trajectories which are defined by|xe(τ)|= exp(−0.18τ).4. CONCLUSIONS AND PERSPECTIVESPolynomial chaos expansions (PCE) constitute aneffective metamodelling technique which has beenused in several practical problems in a wide vari-ety of domains. However, PCE are known to failwhen time-dependent complex systems are consid-ered. The failure is associated with the large dis-similarities between the time-dependent system re-sponses corresponding to different realizations ofthe uncertain parameters.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 20 25 3010−1010−810−610−410−2100τ (s)LOO errors / |x e(τ)|Figure 7: Evolution of the leave-one-out error of PCEin the transformed-time τ0 10 20 30−1−0.500.51τ (s)x(τ) Actual modelPCE prediction0 10 20 30−1−0.500.51τ (s)x(τ) Actual modelPCE predictionFigure 8: Prediction of x(τ) by PCE vs. actual modelsin the transformed time line τ0 10 20 30−1−0.500.51t (s)x(t) Actual modelPCE prediction0 10 20 30−1−0.500.51t (s)x(t) Actual modelPCE predictionFigure 9: Prediction of x(t) by PCE vs. actual modelsin the original time line tWe proposed an approach which consists in rep-resenting the responses into a transformed time linewhere the similarities between different responsetrajectories are maximized. The parameters gov-erning the stochastic time transform are determinedby means of a global optimization problem with anewly introduced objective function that quantifiesthe similarity between two trajectories. The pro-posed approach allows one to solve time-dependentproblems using only low-order PCE. In the consid-ered mechanical example, the approach proves ef-fective in terms of accuracy and computational cost.Further developments are required in order to ap-ply the proposed approach to more complex prob-lems. The formulation in this paper is based onthe assumption that the stochastic responses aremonochromatic. In the case of multi-chromaticsignals (i.e. signals which are richer in frequencycontent), one may decompose the signals intomonochromatic components and then apply a sim-ilar time-transform to each component. One mayalso employs a non-linear time-transform in thesame context. 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