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Computational stochastic mechanicsSession organiser: H. Tanaka, C. Bucher

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Applications of Statistics and Probability in Civil Engineering – Kanda, Takada & Furuta (eds)© 2007 Taylor & Francis Group, London, ISBN 978-0-415-xxxxx-x

Quasi random numbers in stochastic finite element analysis – application toglobal sensitivity analysis

Bruno SudretEDF, R&D Division, Site des Renardières, Moret-sur-Loing, France

Géraud BlatmanEDF, R&D Division, Site des Renardières, Moret-sur-Loing, FranceLaMI, Institut Français de Mécanique Avancée et Université Blaise Pascal,Campus des Cézeaux, Aubière Cedex, France

Marc BerveillerEDF, R&D Division, Site des Renardières, Moret-sur-Loing, France

ABSTRACT: A non-intrusive sensitivity analysis method is described for systems whose uncertain input canbe modeled as random variables. The response of the system is directly represented in the polynomial chaosbasis. Each coefficient of this representation can be expressed as a multi-dimensional integral by a non-intrusiveHilbertian projection. Common simulation schemes, e.g. Monte Carlo Sampling (MCS) or Latin HypercubeSampling (LHS), may be used to estimate the coefficients, with a low convergence rate though. As an alternative,quasi-Monte Carlo (QMC) methods, which make use of quasi-random sequences, are proposed to provide rapidlyconverging estimates. The Sobol’ indices, which measure the global sensitivity of the system to each subset ofinput random variable, are analytically computed from the latter. QMC, MCS and LHS schemes are comparedin three numerical experiments as well as a practical application example, i.e. the sensitivity of the maximaldisplacement of a truss to its members dimensions, material properties and loading.

1 INTRODUCTION

Computer simulations are nowadays commonly usedin structural engineering to accurately model thebehaviour of complex systems. Most of them are deter-ministic and thus provide relevant information as longas the input data is well known, what is seldom thecase in reality.

Probabilistic uncertainty analysis is based on mod-elling uncertainty in the model input by a randomvector X with prescribed joint probability densityfunction, and then characterizing the distribution ofthe random response Y = f (X ). Exploiting this proba-bilistic setting, probabilistic global sensitivity analysisaims at identifying which components in X are themost influential in inducing the uncertainty in Y .In particular, variance-based methods (i.e. ANOVAmethods) aim at decomposing the variance of Y intocontributions related to each input variable. Theseapproaches are reviewed in (Saltelli et al., 2000).Sensitivity indices, e.g. Sobol’ indices (Sobol’, 1993;Saltelli and Sobol’, 1995), are intended to measurethe sensitivities for general models. They are often

estimated using computationally expensive simulationschemes such as Monte Carlo Sampling (MCS) orLatin Hypercube Sampling (LHS).

As an alternative, the Sobol’indices may be straight-forwardly computed from the coefficients of the poly-nomial chaos expansion (PCE) of the random responseY (Sudret, 2006). Such a representation has been usedin the framework of the Spectral stochastic finite ele-ment method (Ghanem and Spanos, 2003), and canbe generalized as a tool for representing functionsof random variables (Soize and Ghanem, 2004). Sev-eral non-intrusive approaches have been proposed inBerveiller (2005) for the computation of the PCE coef-ficients. Most of them are relevant for problems withfew input variables, but they require a number ofnumerical experiments that grows considerably withthe dimension. This curse of dimensionality can bebroken by using MCS methods for computing thecoefficients, at the cost of a low convergence ratethough.

The use of quasi-random numbers (Niederreiter,1992) is proposed in the present paper to providerapidly converging estimates of the PCE coefficients.

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As shown in the sequel, this approach allows to accu-rately compute the Sobol’ sensitivity indices at a lowcomputational cost compared to MCS or LHS.

2 PROBABILISTIC GLOBAL SENSITIVITYANALYSIS

Consider a physical system whose uncertain parame-ters are modeled as independent and uniform randomvariables (X1, . . . , XM ) gathered in the vector X . Thesystem behaviour is described by the following deter-ministic black-box type model (e.g. a finite elementmodel):

where Y denotes the scalar random output.Without loss of generality we assume that the

inputs are uniformly distributed over [0,1]. It has beenshown (Hoeffding, 1948) that the function f can bedecomposed into summands of increasing dimension:

The uniqueness of the decomposition can beobtained by choosing orthogonal summands (Sobol’,1993):

From Eqs. (2),(3) one gets the property:

where I M denotes the unit hypercube [0, 1]M .The combination of the square of Eq.(2) and Eq.(4)

leads to the well-known ANOVA decomposition (Cox,1982):

where the Di1,...,is are partial variances defined as:

The Sobol’sensitivity indices are obtained by divid-ing the latter by the total variance:

The total sensitivity index STk is then defined as thesum of all the sensitivity indices involving the inputparameter Xk :

The relative importance of each input on theresponse variability can thus be investigated by com-paring their indices.

3 POLYNOMIAL CHAOS EXPANSION OF THERESPONSE

3.1 Polynomial chaos expansion

Provided the system response Y has finite variance,the functional f can be represented in an Hilbertianpolynomial basis as follows:

This representation is called finite-dimension poly-nomial chaos expansion (Soize and Ghanem, 2004).The aα’s are unknown deterministic coefficients andthe �α’s are multivariate polynomials which areorthogonal with respect to the joint probability distri-bution of the input vector X , i.e. E[�α(X )�β(X )] = 0for α �= β. As uniform random variables are consid-ered, Legendre M -dimensional polynomials are cho-sen. Note that, to ensure the orthogonality property,the random vector X in Eq. (9) must be uniform over[−1, 1]M , what is trivially obtained by a linear trans-formation of the vector X in Eq. (2). For the sakeof simplicity, the same notation is adopted for bothvectors in the sequel.

The multivariate Legendre polynomials are definedas the tensor product of M unidimensional Legendrepolynomials:

where E[Lαi (U )Lαj (U )] = 0 if i �= j, for any uni-form random variable U over [−1,1].

For computational purposes, the expansion (9) istruncated so that the maximal degree of the polynomi-als does not exceed p:

where |α| = ∑αi and P is the number of terms

given by P = (M+pp

).

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The mean and the variance of the approximatedresponse thus respectively read:

where E[�2

j (X )]

is known analytically.

3.2 PCE-based Sobol’ indices

Let us define by I Pi1, ... ,is the set of the multi-indices α

in (11) such that only the indices (i1, . . . , is) are nonzero:

By rearranging the PCE terms according to theSobol’ decomposition (2), Eq.(11) reads:

The Sobol’ sensitivity indices (7) can thus bederivated as follows:

Note that (7) depends both on the order p of thePCE and on the estimation of the coefficients, whichis detailed in the sequel.

4 NON-INTRUSIVE COMPUTATION OF THEPCE COEFFICIENTS

4.1 Non-intrusive projection scheme

Upon introducing a one-to-one mapping from themulti-indices α to a set of ordered integers j, theresponse PCE (9) can be rewritten as:

Due to the orthogonality of the PCE basis, the pro-jection of the PCE onto the polynomial �j with respectto the scalar product (X , Y ) �→ E[XY ] leads to:

Each PCE coefficient can thus be written as anintegral:

where the expectation E[�2j ] can be derived analyt-

ically, as mentioned for Eq. (13).The multidimensional integral can be computed

using several simulation schemes, as proposed in thesequel.

4.2 Monte Carlo Sampling

Monte Carlo Sampling (MCS) uses independentpseudo-random numbers (x(1), . . . , x(N )) over the unithypercube I M . The integral (9) is approximated by theaverage of the integrand evaluated at the points x(i):

The expected squared error of integration satisfies:

This shows the familiar convergence rate of N −1/2

associated with MCS.

4.3 Latin Hypercube Sampling

The LHS method aims at generating pseudo-randomnumbers with better uniformity over I M than MCS.The support of each input random variable, i.e [0,1] inour context, is divided into N equiprobable intervalsor stratas, leading to a partition of I M in equiprobablesubsets. N realizations of each variable are randomlygenerated by selecting one value in each strata. TheN values obtained for X1 are randomly paired withoutreplacement with the N values obtained for X2. Theresulting N pairs are then randomly combined withthe N realizations of X3, and so on until a set of NM -dimensional samples, i.e. the Latin Hypercubesample, is formed.

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4.4 Quasi-Monte Carlo sequences

As mentioned above, MCS is based on the genera-tion of uniformly distributed pseudo-random numbersover the unit hypercube IM . As an alternative, thequasi-Monte Carlo (QMC) method uses determinis-tic sequences which cover IM better. Such sequencesare called quasi-random or low discrepency sequences(Niederreiter, 1992; Morokoff and Caflisch, 1995).

The unidimensional Halton sequence associated tothe prime number z is defined by expanding the inte-gers 0, 1, . . . , N into base z notation. The n-th term ofthe sequence reads:

where the αi’s are the integers of the base zexpansion of (n − 1):

For instance, for z = 3, the first terms of thesequence read:

It can be observed that this sequence fills I 1 = [0, 1]with terms separated by 1/z and with cycles ofincreasing refinement level z−m.

The M -dimensional Halton sequence is obtainedby pairing M one-dimensional sequences based on Mdifferent primes (z1, . . . , zM ), e.g. the first M primes.Note that in high dimensions, some primes zi can belarge and the number of terms necessary to fill thez−m cycles may be important. Thus the good unifor-mity property of this quasi-random sequence may beaffected.

The Sobol’ sequence circumvents this difficulty bypairing M permutations of one-dimensional Haltonsequences based on z = 2. Nevertheless, with increas-ing dimension, the likelihood of obtaining similarpermutations gets higher, thus leading again to a baduniformity.

As the Sobol’ sequence, the Faure sequence iscompounded of M permutations of one-dimensionalHalton sequences. However the prime z is chosen asthe smallest prime greater than or equal to the dimen-sion M . It is shown that, as long as z ≥ M , an optimalset of permutations can be used. Nonetheless, in highdimensions, the problem of large primes arises.

The uniformity of all these sequences can bemeasured by their discrepancy defined as follows:

where J is an hypercube contained in I M whosesides are parallel to the coordinates axes, and V(J )denotes its volume J .

(x(1), . . . , x(N )

)denotes the N

first terms of the M -dimensional sequence.The discrepancy of the three types of quasi-random

sequences satisfy the relationship:

Provided the integrand f �j has finite mixed deriva-tives (what is a stronger assumption than the finitevariance hypothesis associated to MCS), the Koksma-Hlawka inequality states that:

where ‖·‖BV denotes the so-called variation norm (BVstands for bounded variation).

5 ANALYTICAL TEST EXAMPLES

5.1 Setup of numerical experiments

The MCS, LHS and QMC schemes are used togetherwith 3 test functions with uniform input randomvariables. The models are chosen so that the Sobol’sensitivity indices can be analytically computed. Theestimation accuracy of the second order statisticalmoments and the Sobol’sensitivity indices is observedfor each simulation method. Moreover the globalapproximation accuracy is measured by the lack offit (LOF), which is a normalized Mean Squared Error(MSE) error:

where fN ,P is the P-term truncation of the responsePCE, whose coefficients are estimated using N sam-ples. Let us denote by σ2

YN ,Pthe PCE-based variance

defined in (13) associated with fN ,P .The LOF is estimated in the sequel using

N = 10,000 Monte Carlo samples:

For each method, the results of LOF versus thenumber of samples N are plotted on a log-log scalefor the range Nmin = 10 and Nmax = 10,000. The esti-mated LOF and the estimation errors for N = 10,000are reported.

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Table 1. Polynomial model – relative errors (%) and LOFfor N = 10,000 samples.

MC LHS H S F

µY 1.10 0.35 0.09 0.01 0.03σ2

Y 20.15 9.99 0.81 0.18 0.17S1 0.83 4.33 0.21 0.18 0.27S12 28.67 29.71 1.09 0.05 0.01S123 52.78 215.36 2.00 1.89 0.87

LOF(%) 4.40 4.47 0.04 0.01 0.03

Figure 1. Polynomial model – lack of fit.

5.2 Analytical test examples

5.2.1 Polynomial modelWe first consider the following polynomial function(Sobol, 1993):

where the input variables Xi are uniformly dis-tributed over [0,1].

The numerical application is carried out for M = 3.In this case, the model is polynomial of degree 6. Thusa Legendre PC of order p = 6 is chosen to reproducethe exact model. Therefore the approximation error isonly due to the integration errors while estimating thePCE coefficients. The results for N = 10,000 samplesare reported in Table 1. The rates of convergence ofthe various methods are shown in Figure 1.

5.2.2 Ishigami functionLet us now study the non-monotonic Ishigami function(Homma and Saltelli, 1996) defined as:

Table 2. Ishigami function – relative errors (%) and LOFfor N = 10,000 samples.

MC LHS H S F

µY 1.92 0.29 0.00 0.01 0.02σ2

Y 13.77 4.10 0.18 0.03 0.01

S2 8.67 1.96 0.16 0.03 0.16S13 0.58 6.47 0.13 0.22 0.51ST1 4.23 0.28 0.12 0.03 0.15ST2 0.70 4.16 0.07 0.11 0.43ST3 13.29 4.04 0.51 0.04 0.06

LOF(%) 5.78 4.30 0.14 0.10 0.15

Figure 2. Ishigami function – lack of fit.

where the input variables Xi are uniformly dis-tributed over [ − π, π].

The application example is carried out using thenumerical values a = 7 and b = 0.1. A 9th order PCEis used. Note that as the model is non-polynomial, thisPCE truncation leads to a metamodelling error whichis added to the coefficients estimation error.The resultsare reported in Table 2 and the rates of convergence areplotted in Figure 2.

5.2.3 Sobol’ functionLet us finally consider the so-called Sobol’ function(Sobol, 2003):

where the input variables Xi are uniformly dis-tributed over [0,1] and the ai’s are non negative. Forthis numerical application, q = 8 is chosen togetherwith a = (1, 2, 5, 10, 20, 50, 100, 500). A second orderPCE is used. The results are reported in Table 3 andthe rates of convergence are shown in Figure 3.

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Table 3. Sobol’ function – relative errors (%) and LOF forN = 10,000 samples.

MC LHS H S F

µY 0.55 0.07 0.01 0.01 0.10σ2

Y 11.06 4.89 9.80 9.86 10.64ST1 2.21 3.71 1.32 1.15 0.10ST2 9.59 4.07 5.85 5.97 10.11ST3 18.88 2.49 7.84 8.60 3.38ST4 39.15 4.45 10.37 9.20 17.41

LOF (%) 17.37 12.26 9.59 9.58 10.84

101 102 103 10410-2

10-1

100

101

102

103

Number of samples

Lack

of f

it

MCLHSHSF

1

Figure 3. Sobol’ function – Lack of fit.

5.3 Discussion

Figures 1 and 2 clearly show that the QMC sequencesconverge more rapidly than the MCS and LHS meth-ods in the two 3-dimensional examples. Moreover, theQMC sequences have significantly lower LOF andlead nearly to the same level of global accuracy withN = 1,000 samples as MCS and LHS with N = 10,000numerical experiments. Tables 1 and 2 show that theQMC sequences provide more accurate estimators ofthe statistical moments and the Sobol’ indices. Notethat the Sobol’ sequence performs globally slightlybetter than the Halton and the Faure sequences.

Figure 3 displays the superiority of the Sobol’ andthe Halton sequences in term of convergence rate forthe 8-dimensional test. The Faure sequence has notice-ably higher LOF than MCS and LHS, although itconverges as fast as the other QMC sequences. Onthe other hand, a LOF lower bound of about 10−1 isobserved for each method. This limit is due to a toosevere truncation of the response PCE (second order).Nonetheless, the Sobol’ indices are rather correctlyestimated, with relative errors rarely greater than themodelling error.

Figure 4. Truss structure with 23 members.

Table 4. Truss example – Variation ranges of the inputvariables (uniform distributions).

Variable Lower bound Upper bound

E1, E2 (Pa) 1.995 × 1011 2.205 × 1011

A1 (m2) 1.9 × 10−3 2.1 × 10−3

A2 (m2) 0.9 × 10−3 1.1 × 10−3

P1–P6 (N) 3.5 × 10−3 6.5 × 10−3

6 GLOBAL SENSITIVITY ANALYSIS OF ATRUSS STRUCTURE

6.1 Setup of the truss example

Let us study the truss in Figure 4. A model ofthe structure is built using EDF’s finite elementcode CODE_ASTER (http://www.code-aster.org). It ismade of 23 bar elements.

Ten independent input variables are considered,whose variation ranges are reported in Table 4. Thedisplacement v1 is approximated by a 2nd order Leg-endre PCE. The Sobol’ sequence, which performedconstantly well in the analytic examples, is comparedto MCS for computing the PCE coefficients. Theresults in terms of mean, standard deviation and Sobol’indices of the response are given in next section.

6.2 Results

Figure 5 shows that the Sobol’ sequence provides amean estimate which has reached a disprepancy lessthan 0.3 % from N = 1,000 samples, whereas slightdiscrepancies are still observed for the MCS estimatefor N ≥ 2,000. Moreover, the former method resultsin a standard deviation estimate that has converged(i.e. with an absolute discrepancy ≤ 2.5 × 10−2) fromN = 500 samples, whereas the latter has still notreached such an accuracy at N = 3,000. Lastly, signif-icants fluctuations in the MCS estimates of the Sobol’indices are observed in Figure 7, leading to unreliableresults.

This poor performance of MCS is confirmed byFigure 8. Indeed, it can be observed that this method

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0 500 1000 1500 2000 2500 30000.0584

0.0586

0.0588

0.059

0.0592

0.0594

0.0596

0.0598

0.06

Number of samples

MCS

Figure 5. Estimates of the displacement mean.

0 500 1000 1500 2000 2500 30000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Res

pons

e st

anda

rd d

evia

tion

Number of samples

MCS

Figure 6. Estimates of the displacement standard deviation.

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Number of samples

Sob

ol’ i

ndic

es e

stim

ates

SP5

MCS

P5 S

SA1

MCS

A1 S

Figure 7. Estimates of the Sobol’ indices associated to P5and A1.

E1 E2 A1 A2 P1 P2 P3 P4 P5 P60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Input variables

Tot

al S

obol

’ ind

ices

MCS

Figure 8. Total Sobol’ indices using N = 3,000 samples.

yields estimates of the total Sobol’ indices which doesnot reflect at all the symmetry of the problem. In con-trast, the Sobol’ sequence gives similar importance tothe forces that are symmetrically located (e.g. P3 andP4). Furthermore it logically leads to higher sensitiv-ity indices associated to the forces which are closeto the midspan of the structure than to those locatedat the ends. Finally, the relative values of the Sobol’indices associated to variables E1, E2, A1 and A2 arephysically meaningful when computed by QMC (thecharacteristics E1 and A1 of the horizontal bars aremore influential on the displacement v1 than E2 andA2) whereas the results obtained by MCS are in theopposite order.

7 CONCLUSION

The quasi-Monte Carlo method have been proposed toprovide rapidly converging estimates of the responsePCE coefficients, thus allowing an efficient computa-tion of the Sobol’ sensitivity indices. This method isbased on the generation of deterministic quasi-randomsequences which ensure a better coverage of the unithypercube than the pseudo-random numbers used inMonte Carlo schemes.

Three quasi-random sequences were used, namelythe Halton, the Sobol’ and the Faure sequences. Theyprovided similar results in two 3-dimensional analyti-cal examples, overperforming MCS and LHS in termsof convergence rate and estimation accuracy. How-ever, the Faure sequence behaved noticeably poorly ina 8-dimensional numerical application test, whereasthe other sequences kept their advantage. The Sobol’sequence performed globally better in all the numeri-cal experiments. Further, it confirmed its superiority toMCS in the example of a truss finite element model.In the lack of further knowledge, this sequence will

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be thus used as our default simulation scheme whencoping with uniform measures.

While recognizing that the choice of uniform ran-dom variables allows to measure the global sensitivityof the deterministic model from a pure exploratoryviewpoint, one must admit that selecting nonuniformdistributions should often be preferred to properly rep-resent the natural variability of the inputs (e.g. lognor-mal distributions for some material properties). Thedirect generalization of the quasi-Monte Carlo meth-ods to measures with unbounded support might not bestraightforward, since the necessary changes of vari-ables lead to functions with infinite variation norms.However, the recently developped quasi-ImportanceSampling approach (Hörmann and Leydold, 2005)seems to be a simple way to overcome this difficulty.Good performance of the QMC methods in problemswith nonuniform input random variables is thereforeexpected.

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Hoeffding,W. (1948).A class of statistics with asymptoticallynormal distributions. Ann. Math. Stat. 19, 293–325.

Homma, T. and A. Saltelli (1996). Importance measures inglobal sensitivity analysis of non linear models. Rel. Eng.Syst. Safe. 52, 1–17.

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