An Efficient Sampling Method for Regression-Based Polynomial Chaos Expansion

Transcription

1 Commun. Comput. Phys. do: /ccp a Vol. 13, No. 4, pp Aprl 2013 An Effcent Samplng Method for Regresson-Based Polynomal Chaos Expanson Samh Zen 1,, Benoît Colson 1 and Franços Glneur 2 1 Samtech H.Q., LMS Internatonal, 8 rue des chasseurs ardennas Angleur, Belgum. 2 Center for Operatons Research and Econometrcs & Informaton and Communcaton Technologes, Electroncs and Appled Mathematcs Insttute, Unversté catholque de Louvan, B-1348 Louvan-la-Neuve, Belgum. Receved 2 September 2011; Accepted (n revsed verson) 20 Aprl 2012 Avalable onlne 21 September 2012 Abstract. The polynomal chaos expanson (PCE) s an effcent numercal method for performng a relablty analyss. It relates the output of a nonlnear system wth the uncertanty n ts nput parameters usng a multdmensonal polynomal approxmaton (the so-called PCE). Numercally, such an approxmaton can be obtaned by usng a regresson method wth a sutable desgn of experments. The cost of ths approxmaton depends on the sze of the desgn of experments. If the desgn of experments s large and the system s modeled wth a computatonally expensve FEA (Fnte Element Analyss) model, the PCE approxmaton becomes unfeasble. The am of ths work s to propose an algorthm that generates effcently a desgn of experments of a sze defned by the user, n order to make the PCE approxmaton computatonally feasble. It s an optmzaton algorthm that seeks to fnd the best desgn of experments n the D-optmal sense for the PCE. Ths algorthm s a couplng between genetc algorthms and the Fedorov exchange algorthm. The effcency of our approach n terms of accuracy and computatonal tme reducton s compared wth other exstng methods n the case of analytcal functons and fnte element based functons. AMS subject classfcatons: 60H15, 62K20, 62K05 Key words: Polynomal chaos expanson, regresson, D-optmal desgn, Fedorov Algorthm, genetc algorthms. 1 Introducton In the recent years, there has been an ncreasng nterest n the smulaton of systems wth uncertantes. Due to the uncertantes n the nput parameters, the response of the mechancal system has a determnstc component and a random component; ths random Correspondng author. Emal addresses: samh.zen@lmsntl.com (S. Zen), benot.colson@lmsntl.com (B. Colson),Francos.Glneur@uclouvan.be (F. Glneur) c 2013 Global-Scence Press

2 1174 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp component s often gnored n tradtonal engneerng practce. The relablty analyss conssts n defnng the probablty densty functons (pdf) of the uncertan nput parameters and then propagatng them through the mathematcal model of the mechancal system n order to characterze the random component of the output. The polynomal chaos expanson (PCE) method bulds a multdmensonal polynomal functon that approxmates the output of the system around ts nomnal value wth respect to the uncertanty of the nput parameters. There are two non-ntrusve methods to construct the PCE approxmaton: the projecton method and the regresson method (see [22] and [23]). Both of them are black box methods that requre a set of ndependent smulatons for dfferent values of the nput parameters. The total computatonal tme depends on three factors: the computer resources, the computatonal tme of one fnte element analyss (FEA) and the number of smulatons needed to buld PCE approxmaton. Usually, the ndustral applcatons requre expensve fnte element analyss. Hence, f the number of analyses for buldng the PCE s large, the total computatonal tme becomes unfeasble. Reducng the total computatonal tme by tunng the frst two factors s not an easy task. The computer resources are usually lmted and reducng the computatonal tme of one FEA means reducng ts accuracy. Therefore, one has to reduce the thrd factor, the number of analyses for the PCE, n order to be able to carry out a relablty analyss of an expensve FEA-based applcaton. Recently, some new methods have been ntroduced to reduce the number of ndependent smulatons. The projecton method requres runnng a set of smulatons defned over a sparse grd of nodes followng some quadrature rules. In [16 19], the reducton of the sze of sparse grd s based on the ansotropy of the problem and the contrbuton of each node of the sparse grd n the overall accuracy of the numercal evaluaton. The regresson method requres the defnton of a desgn of experments dependng on the PCE polynomal functon. In [1, 20, 21], a model reducton approach s consdered where some terms of the PCE approxmaton are dscarded accordng to ther pertnence n the model. Ths leads to a reducton of the sze of the desgn of experments. The precedng methods prescrbe ther number smulatons wthout provdng to the user a drect control over the total number of smulatons to perform the PCE approxmaton. The contrbuton of ths paper s to propose a samplng algorthm for the regresson method where the user can choose the number smulatons dependng on the avalable computer resources, the computatonal tme of one FEA and the desred total computatonal tme. The proposed algorthm gves the possblty to carry out a PCE approxmaton wth the fewest number of smulatons whch s equal to the number of terms n the PCE. Therefore, wth ths approach t s always possble to carry out a relablty analyss based on an expensve FEA model and a large number of uncertan parameters. The accuracy of the PCE approxmaton gven by the regresson method s proportonal to the determnant of the regresson matrx. The approach proposed n ths paper conssts n maxmzng ths determnant for a gven number of samples. Ths leads to a small szed desgn of experments that covers effcently all the space of the uncertan-

3 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp tes. The proposed algorthm for maxmzng the determnant of the regresson matrx s a couplng between genetc algorthms and the Fedorov exchange algorthm. The motvaton for ths choce s dscussed n Secton 4. The effcency of our approach n terms of accuracy and computatonal tme reducton s compared wth other exstng methods (the projecton method) for analytcal and ndustral cases. 2 Probablty space and random varables Computng the output of a mechancal system wth uncertan parameters s consdered as a random experment and can be studed usng the PCE. In order to gve a mathematcal framework of the PCE, a bref revew of some defntons regardng probablty space and random varables s ntroduced n the sequel. A probablty space s a mathematcal entty whch models the uncertanty n the nput and the output of the mechancal system. Three ngredents (Ω,F, P) are necessary to defne a probablty space, where Ω s the set of all possble outcomes, F s a σ-algebra over Ω, contanng all possble events and P s a functon F R whch gves a probablty measure for each random event. A random varable X s a functon X :(Ω,F, P) R. Usng the precedng defntons, one can also defne the expectaton and the varance of a random varable: X= E[X]= XdP, (2.1) var(x)=e[(x X) 2 ]= (X X) 2 dp. (2.2) We call < X,Y>= XYdP the nner product of two random varables on the probablty space and L 2 (Ω,F,P) the set of random varables havng a fnte varance. Before ntroducng the PCE, one must also have a look at the Hermte polynomals. These polynomals are defned n a recursve way as follows: H 0 = 1, (2.3) H n+1 (x)= xh n (x) (n)h n 1 (x), (2.4) and they have the property of beng orthogonal wth respect to the Gaussan measure. If H and H j are two Hermte polynomals of degrees and j respectvely, we have: H (x)h j (x)e x2 /2 dx=!δ j. (2.5) Thus, f ξ s a Gaussan random varable, the two random varables H (ξ) and H j (ξ) are orthogonal wth respect to the above nner product and the Gaussan measure. Ths orthogonalty property consttutes the bass of the PCE. Note that, f ξ s a unform random varable, the Hermte polynomals must be replaced by the Legendre polynomals and the densty functon e x2 /2 by 1.

4 1176 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp The polynomal chaos expanson (PCE) In the followng, the bass of the regresson method and the D-optmal crteron are gven. More detals can be found n [7]. The PCE method conssts n consderng a set of orthogonal multvarate polynomals wth respect to some nner product. Let M be the number of uncertan parameters n the mechancal system and{ξ (ω)} =1 M be a set of M ndependent Gaussan random varables. Each random varable s assocated to an uncertan parameter and represents ts perturbaton. Consder Γ p the space of multvarate polynomals of degree less or equal to p and Γ p the space of polynomals of degree equal to p and orthogonal to Γ p 1. We have Γ p = p Γ =1. If the nner product s consdered wth the Gaussan measure, Γ p s spanned wth the multvarate Hermte polynomals {ψ α (ξ)} of degree p, where α N M and α ==1 M α =p. These multvarate Hermte polynomals{ψ α (ξ)} are defned as the product of M unvarate Hermte polynomals of degree α : ψ α (ξ)= M =1 H α (ξ ). (3.1) If S denotes the random output of the mechancal system and has a fnte varance, t can be expressed as an nfnte seres of multvarate Hermte polynomals (see [14]): α = S= α =0 S pce α ψ α (ξ)= =0 S pce ψ (ξ). (3.2) Here, the mult-ndex α of ψ s replaced by an ndex gven the unvocal relatonshp between these two notatons. S pce are the determnstc coeffcents of the PCE. One of the advantages of such a representaton of the random output S s that computng the expectaton and varance of S s straghtforward. From the orthogonalty property of ψ, one can deduce that: S=S pce 0, (3.3) var(s) = (S pce =1 ) 2. (3.4) For computatonal reasons, ths nfnte seres must be truncated to degree p and S s replaced by S wth: S= N =0 S pce ψ (ξ), (3.5) where N= (M+p)! M!p! 1 s the number of terms nvolved n S. In ths paper, we consder the regresson method to compute the coeffcents S pce of the PCE. Ths method conssts n defnng a desgn of experments, Ξ={ξ j }j=1,,q, whch

5 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp s a set of Q dfferent perturbatons of the nput parameters and then computng (by smulaton) the output of S for each value ξ j of the desgn of experment. These output values are gathered n a vector S sm R Q. The coeffcents S pce are then computed by solvng the followng lnear least squares problem: mn R= 1 S pce 2 Q j=1 ( S sm j N =0 S pce ψ (ξ j )) 2, (3.6) where R s the squared resdual between the smulated and the approxmated output values. The soluton of ths optmzaton problem s derved n the followng. Wrtng the dervatve of R wth respect to S pce k equal to zero leads to: Q j=1 ( ψ k (ξ j ) S sm j N =0 ) S pce ψ (ξ j ) = 0 k=1,,n. (3.7) The above equaton can be rearranged n the followng form (called the normal equatons): Q N j=1=0 ψ k (ξ j )ψ (ξ j )S pce Q = j=1 ψ k (ξ j )S sm j k=1,,n. (3.8) Let Ψ j = ψ (ξ j ) be a rectangular matrx n R Q,N. Each row j of ths matrx s the set of all ψ s computed for the experment ξ j. Hence, the matrx representaton s deduced from the above equatons: S pce =(Ψ t Ψ) 1 Ψ t S sm. (3.9) Ψ t Ψ s called the nformaton matrx [7]. 3.1 D-optmal desgn In ths secton, t s shown how the choce of the DOE Ξ={ξ j } j=1,,q affects the accuracy of the estmaton of S pce. We recall that the output of the mechancal system s expressed as a seres of multvarate Hermte polynomal truncated at degree p (Eq. 3.5): S sm = N =0 S pce ψ (ξ)+ε, (3.10) where ε s the truncaton error between the PCE and S sm and s a random varable. The presence of ths random truncaton error nduces a random error n the estmaton of S pce. The varance of S pce s derved as follows: var(s pce )=var((ψ t Ψ) 1 Ψ t S sm ) =(Ψ t Ψ) 1 Ψ t var(s sm )((Ψ t Ψ) 1 Ψ t ) t. (3.11)

6 1178 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp Assumng that var(s sm )= Iσ 2 meanng that the truncaton errors n each smulaton are ndependent and have a constant varance equal to σ, we obtan: var(s pce )=(Ψ t Ψ) 1 σ. (3.12) From Eqs. (3.9) and (3.12), one can deduce the relatonshp between Ξ and var(s pce ). Each S pce s a lnear combnaton of the terms of S sm. Usng the central lmt theorem, S pce s a vector of Gaussan varables of varance (Ψ t Ψ) 1 σ. It s known that ts confdence regon s a hyper-ellpsod of volume proportonal to the determnant of (Ψ t Ψ) 1. In order to reduce var(s pce ), the error n the estmaton of S pce, one needs to reduce the confdence regon. A D-optmal desgn s a desgn that gves an estmaton of S pce wth the smallest confdence regon. It s the soluton of the followng optmzaton problem: max Ξ det(ψt Ψ). (3.13) Ths optmzaton problem consttutes the man task of our work: fndng the D-optmal desgn n the case of the regresson based PCE. Several methods based on Fedorov exchange formula or heurstc optmzaton have been proposed to solve ths partcular optmzaton problem. In the followng secton, we gve a revew of these methods and we ntroduce our couplng optmzaton algorthm adapted to the case of regresson based PCE as was dscussed n Secton 1. 4 Optmzaton algorthms We recall that the purpose of ths optmzaton algorthm s to generate a desgn of experments for the PCE approxmaton, as much accurate as possble, gven a user defned number of experments. Each uncertanty of each parameter s dscretzed such that t takes a fnte number of values, dependng on the degree of the PCE. The combnaton of the dscrete values of all the parameters consttutes a grd of nodes where each node s an experment canddate. The desgn of experments s a set of experments chosen from ths grd of nodes. One can see that the sze of the grd ncreases exponentally wth the number of uncertan parameters and the degree of the PCE. A common crteron for selectng a DOE s the D-optmalty one by maxmzng the determnant of the nformaton matrx. Ths crteron can be computed usng the Fedorov algorthm (see [4]), whch belongs to the famly of exchange algorthms. It s detaled n Secton 4.1. The underlyng dea s to fnd the best DOE from a grd of nodes. At each teraton, an experment of the DOE s compared wth all the experments canddates of the grd by applyng the Fedorov exchange formula and t s exchanged wth the best one. The algorthm can handle effcently a DOE of large sze but t can become expensve n computatons when the sze of the grd s large. Moreover, t can be trapped nto a local optmum.

7 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp Two dfferent approaches have been proposed to optmze the determnant of the nformaton matrx wthout scannng all the experments canddates. In [2, 3] a genetc algorthm s used to optmze the D-optmalty crteron and n [5, 6] a heurstc search s preferred. These algorthms are global methods and they can be a good choce over the Fedorov method because they do not scan the whole set of canddates at each teraton. However, the crossover operator presents two major drawbacks. Frst, t can be expensve when DOEs have a large sze because t requres the computaton of the determnant of the nformaton matrx. Second, t has been observed n the numercal examples n Secton 5 that the crossover operator fals to optmze the determnant when the set of experments canddates s also large. For ths reason, an algorthm s proposed that couples the Fedorov exchange algorthm wth the genetc algorthms to overcome ths dffculty. The use of the genetc algorthms allows to avod scannng all the experments canddates. A cheap crossover can be defned usng the Fedorov exchange formula whch avods computng the determnant of the nformaton matrx. Some numercal experments are carred out to show the performances of ths couplng wth large DOEs. 4.1 The Fedorov algorthm We start by showng how the determnant of Ψ t Ψ changes when an experment of the DOE s exchanged wth a new one. The matrx Ψ t Ψ can be wrtten as Q j=1 ψ(ξj )ψ(ξ j ) t where ψ(ξ j )={ψ (ξ j )} =1,,N s the row of the matrx Ψ correspondng to experment j. Ψ t Ψ s a sum of outer products over the experments of the DOE. When a new ξ Q+1 experment s added to the DOE, the new nformaton matrx Ψ t new Ψ new becomes Q+1 j=1 ψ(ξj )ψ(ξ j ) t. From the matrx determnant lemma, the determnant of Ψ t new Ψ new can be updated from the one of Ψ t Ψ as follows: det(ψ t newψ new )=det(ψ t Ψ+ψ(ξ k )ψ(ξ k ) t ) ( ) = det(ψ t Ψ) 1+ψ(ξ k ) t (Ψ t Ψ) 1 ψ(ξ k ). (4.1) If an experment k s removed from the DOE, the update formula s the same as above but the sgn+s replaced wth a. Now, when an experment l from the DOE s exchanged wth an experment k, the determnant of the new nformaton matrx Ψ t new Ψ new can be obtaned by applyng the above formula twce: ( ) det(ψ t newψ new )=det(ψ t Ψ) 1+ (ξ k,ξ l ), (4.2) where (ξ k,ξ l )=d(ξ k,ξ k ) d(ξ k,ξ k )+d 2 (ξ k,ξ l ) d(ξ k,ξ k )d(ξ l,ξ l ), (4.3) d(ξ k,ξ l )=ψ(ξ k ) t (Ψ t Ψ) 1 ψ(ξ l ). (4.4)

8 1180 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp When two experments are exchanged, the determnant of the nformaton matrx changes wth rate. The Fedorov algorthm for maxmzng det(ψ t Ψ) s based on the above formula. Consder a set C of experment canddates (a grd of nodes) and Ξ C a random DOE of Q experments. The dea of the algorthm s that for each experment ξ l Ξ we compute the values of the functon (ξ k,ξ l ) for all ξ k C\Ξ and exchange ξ l wth the ξ k that gves the maxmal value of. It can be summarzed n Algorthm 4.1. One can see that ths algorthm s effcent n the sense that t uses the update formula to compute the determnant at each teraton whch s not an expensve operaton. However, t can stll be qute expensve when the sze of C becomes large. It s mportant to note that the ntal DOE has to be nonsngular, otherwse the determnant of the matrx wll stay equal to zero durng all the executon of the algorthm. Algorthm 4.1: The Fedorov Algorthm Generate a random Ξ C Compute the determnant of Ξ for all ξ j Ξ do set max =0 for all ξ l C\Ξ do Compute (ξ k,ξ l ) f (ξ k,ξ l )> max then Set max = (ξ k,ξ l ) Exchange ξ l wth ξ k Update the determnant of Ξ end f end for end for 4.2 Generatng a nonsngular random DOE If a DOE s generated by randomly selectng some experments from a set of canddates and the sze of the DOE s small, there s a hgh probablty to get a sngular DOE. In ths case, the Fedorov exchange algorthm s not applcable because the determnant wll reman zero durng the whole procedure. In the followng, a method s proposed whch ensures that t always generates a nonsngular DOE. We suppose that each parameter uncertanty can take p+1 dstnct values,{r 0,r 1,,r p }, ndexed from 0 to p. An experment s a node of the grd {r 0,r 1,,r p } M and each experment s dentfed by an mult-ndex α N M. The set of experments such that α p guarantees to have a nonsngular DOE. Takng a DOE wth these nodes and followng the Fedorov exchange formula, each experment of ths DOE can be exchanged wth a randomly selected one from the grd f the determnant of the DOE ncreases. Some hghlghts of the proof are gven:

9 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp We recall that the PCE approxmaton s expressed n a Hermte bass. The terms of the approxmaton can be developed n the canoncal bass whch leads to a polynomal expresson: α =p S= α =0 S pce α α =p ψ α (ξ)= α =0 S can α φ α (ξ), (4.5) where the φ α (ξ) are the canoncal monomals, Sα can are the coeffcents n ths bass and α s a mult-ndex n N M. Ths notaton wth the mult-ndex s equvalent to the one n (3.5). The change of bass s defned n a matrx form, S pce =C.S can where C s a nonsngular matrx. Ths mples that one can drectly get the Sα pce from the Sα can. If the PCE s of degree p, t s easy to see that the Sα can assocated to the monomals α S of degree p are equal to ξ / M α =1 α!. the α-dervatve of a polynomal functon of degree p wth α =p s equal to ts fnte dfference approxmaton because the fnte dfference approxmaton of order α s derved from the Taylor expanson of the functon and the Taylor expanson of order α of a polynomal functon s equal tself. Thus, one can choose the experments of the DOE such that they correspond to a fnte dfferences scheme and drectly compute the coeffcents Sα can wth α = p. Once these coeffcents are evaluated, ths operaton can be repeated teratvely for the degrees from p 1 to 0 and hence all the coeffcents are computed. Gven that the set of nodes wth α p contans all the fnte dfferences schemes needed for computng all the PCE coeffcents, a DOE contanng these nodes s nonsngular. 4.3 The coupled Fedorov-genetc algorthm In [2, 3, 12], genetc algorthms have been used for fndng D-optmal desgns. They are based on the defnton of a random populaton whch n ths case s a set of DOE canddates. These ndvduals wll nteract wth each other n order to evolve toward the optmal soluton. Ths evoluton process can be summarzed n the followng steps. Frst, the populaton of DOEs s ntalzed by generatng randomly a certan number of DOEs. Next and at each teraton, the determnant of each DOE of the populaton s computed. A random procedure, called crossover, s appled to select the parents accordng to ther determnant and obtan the new DOEs offsprng of the new populaton. For more nformaton on genetc algorthms see [11, 15]. Durng the optmzaton procedure, all the DOEs are nonsngular. Ths s guaranteed by couplng the Fedorov exchange formula wth the genetc algorthms. In the ntalzaton step, the random and nonsngular DOEs are generated lke n the prevous secton. In the crossover step, the two parents DOEs exchange some of ther

10 1182 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp experments wthout producng new sngular DOEs. Let the two parents DOEs be Ξ 1 and Ξ 2 and N c be the number of crossover ponts whch s the number experments to be exchanged between them. An experment s randomly selected n Ξ 1 and t s exchanged wth another one n Ξ 2 such that accordng to the Fedorov exchange formula the determnant of both DOEs does not decrease. Ths operaton s repeated N c tmes to generate new offsprngs. Algorthm 4.2: The GA-Fedorov Algorthm Intalze a random populaton {Ξ } of DOEs for k=1 to number of teratons do Compute all the determnants of the populaton {Ξ } for l=1 to populaton sze do Select two parents randomly accordng to ther determnants Reproduce an offsprng usng the Fedorov algorthm: for m=1 to number of crossover ponts do Choose randomly an experment ξ of the offsprng Compute (ξ,ξ j ) for all ξ j n the second parent Replace ξ wth the ξ j that has a maxmal end for end for Replace the parents populaton {Ξ } wth the offsprng populaton end for 5 Numercal results In ths secton, we show the effcency of the couplng algorthm n generatng accurate and small-szed DOE for the regresson based PCE. Three sets of tests are performed: a comparson wth the standard GA algorthm, a comparson wth the closest to the orgn samplng method proposed n [1] and fnally a comparson wth the projecton method. 5.1 Comparson wth the Standard GA Algorthm We start by comparng the standard genetc algorthm wth the Fedorov-GA algorthm. The set C of experment canddates s taken the same as the one n [1]. It s the grd of nodes {r 0,r 1,,r p } M where r are the roots of the Hermte polynomals of degree p+1. The number of experments n each DOE s taken as twce the number of coeffcents n the PCE. We compare the effcency of our Fedorov based crossover wth the standard one. The PCE s of degree 4 and has 6 varables. The number of experments n the DOEs s 420. In Fgs. 1 and 2, t s shown how the standard crossover s not able to mprove the determnant of the DOE unlke the Fedorov based crossover of our algorthm. In the frst case the determnant decreases durng the teratons, whle n the case of the

11 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp Populaton Sze= 10, #Crossover Ponts=30 Populaton Sze= 20, #Crossover Ponts=30 Populaton Sze= 20, #Crossover Ponts= Log of determnant Number of teratons Fgure 1: Standard genetc algorthm: case of Hermte polynomal of degree p=4, wth M=6 varables and N = 420 experments Log of determnant Populaton Sze= 10, #Crossover Ponts=30 Populaton Sze= 20, #Crossover Ponts=30 Populaton Sze= 20, #Crossover Ponts= Number of teratons Fgure 2: Coupled Fedorov genetc algorthm: case of Hermte polynomal of degree p=4, wth M=6 varables and N= 420 experments. coupled algorthm the determnant ncreases. The optmzaton experments are repeated for several populaton szes (10 and 20 ndvduals) and several number of crossover ponts (N c = 30 and 50). 5.2 Comparson wth the closest to the orgn samplng method In the second experment, we compare the samplng method of DOE proposed n [1] wth our couplng method. The DOE n the quoted paper s made of the N closest ponts to the orgn n C. Ths method, as t s mentoned n the paper, has the drawback of requrng

12 1184 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp Table 1: Comparng the mnmal sze of the DOE requred by the closest to orgn method and the Fedorov-GA method. p M # of experments /det(ψ t Ψ) N+1/det(Ψ t Ψ) 2 N/det(Ψ t Ψ) (closest to orgn) (Fedorov-GA) (Fedorov-GA) /29 36/35 70 / /72 57/71 112/ /145 85/ / /22 71/ / / / / / / /551 a large number of experments n order to have a nonsngular DOE. Table 1 shows the smallest DOE szes that both methods can generate and t compares ther accuraces by computng ther respectve determnants. The mnmal number of experments requred for the couplng algorthm s always equal to the number of terms n the PCE. It can be seen that the closest to orgn method requres much more experments than Fedorov-GA. Ths can be very expensve n computatons. The determnant of the DOE wth the closest to orgn method s n the same range of values as the one obtaned wth the Fedorov-GA. In concluson, the coupled Fedorov-GA s more effcent because t gves DOE wth the same accuracy and much less samples. 5.3 Comparson wth the projecton method In ths thrd set of tests, the couplng algorthm s compared to the projecton method n terms of cost and accuracy. Three tests are based on analytcal functons and one on a fnte elements model. The exact standard devatons of the analytcal functons are known n advance. By comparng t to the computed standard devaton, ths gves the accuracy of the PCE approxmatons by both methods. Test one: f(x)= 3 =1 sn(x ), x [ π,π] and the varance of f s equal to 1.5. Test two: g(x)=sn(x 1 )+a.sn 2 (x 2 )+b.x 4 3 sn(x 1), x [ π,π] and the varance of g s equal to a2 8 π4 π8 +b 5 +b In ths example, a=7 and b=0.1. Test three: h(x)= 3 4x 2 +a =1 1+a, wth a=(1,2,5) and x [0,1]; the varance s equal to Table 2 gves the results correspondng to the analytcal functons. It also shows the relatve errors of the two methods n computng the standard devatons of the three analytcal functons. One can see that both methods are able to compute accurately the standard devatons of these functons: the hghest relatve error s only 1.23% wth the Fedorov-GA method for test 2. Ths method s also 2.5 to 3 tmes faster than the projec-

13 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp Dstrbuted Force Strnger Panel 1 Panel Clamped Surface Fgure 3: Descrpton of the fnte element based test: the sold arrows are the uncertan varables. Fgure4: Thefnteelementmodelofthestructure: two panels and one strnger. Fgure 5: The frst bucklng mode of the structure. Fgure 6: The second bucklng mode of the structure. Fgure 7: The thrd bucklngmode of thestructure.

14 1186 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp Table 2: Comparson of the results obtaned by the projecton method and the coupled Fedorov-GA method for the analytcal functons. Projecton Method Fedorov-GA Method # samples StD Rel. Err. # samples StD Rel. Err. f(x) % % g(x) % % h(x) % % Table 3: Comparson of the means and the standard devatons obtaned wth the projecton method and the coupled Fedorov-GA method for the fnte element test case. Projecton Method Fedorov-GA Method Rel. Dff. Mean StD Mean StD Mean StD Degree=2 (78 samples) Degree=2 (30 samples) Weght % 0.7% Mode % 6.1% Mode % 2.7% Mode % 3.5% Degree=3 (257 samples) Degree=3 (84 samples) Weght % 1.1% Mode % 6.4% Mode % 0% Mode % 1.1% ton method. Ths result shows that the Fedorov-GA method s able to generate a PCE approxmaton that accurately approxmates these functons and wth a low cost. Table 3 gves the results correspondng to the fnte element based functons. Here, the exact means and standard devatons are not known n advance. Hence, the relatve dfference between the standard devatons of both methods s compared. One can see that both methods gve close results because the hghest relatve dfference s about 7% and t s n the case of mode 1 wth degree two. On can see that the Fedorov-GA method gves comparable results to the projecton method wth a computatonal cost 2.5 to 3 tmes lower. Ths concludes that these fnte element based functons are accurately approxmated wth the PCE generated by the Fedorov-GA method. 6 Concluson In ths paper an optmzaton algorthm s proposed for generatng small szed DOEs for regresson based polynomal chaos expanson. Ths method has the advantage of not prescrbng the number of samples but t gves the possblty to the user to defne the number of samples. For example, one can choose to take the fewest number of samples whch s the number of terms nvolved n the PCE. The optmzaton algorthm s a ge-

15 S. Zen, B. Colson and F. Glneur / Commun. Comput. Phys., 13 (2013), pp netc algorthm wth a crossover operator based on the Fedorov algorthm. It s used to maxmze the D-optmal crteron of the DOE. We show the effcency of ths coupled algorthm by comparng t wth the standard GA, the closest to orgn samplng method and the projecton method. Some analytcal and fnte element based tests are consdered. The effcency of the proposed algorthm s shown n terms of cost reducton and accuracy n all the precedng cases. Acknowledgments The research leadng to these results has receved fundng from the Walloon regon of Belgum, conventon number 5856, subventon FIRST-ENTREPRISE. Ths text presents results of the Belgan Program on Interunversty Poles of Attracton ntated by the Belgan State, Prme Mnster s Offce, Scence Polcy Programmng. The scentfc responsblty rests wth ts authors. References [1] B. Sudret, Global senstvty analyss usng polynomal chaos expansons, Relab. Eng. Syst. Safe., 93, pp (2008). [2] J. Poland, A. Mtterer, K. Knödler, and A. Zell, Genetc algorthms can mprove the constructon of D-optmal expermental desgns, Advances n Fuzzy Systems and Evolutonary Computaton, pages (2001). [3] D. Dran, W. Matthew Carlyle, D. C. Montgomery, C. Borror and C. Anderson-Cook, A Genetc Algorthm Hybrd for Constructng Optmal Response Surface Desgns, Qualty and Relablty Engneerng Internatonal, 20, pp (2004). [4] N. Nguyen and A. J. Mller, A revew of some exchange algorthms for constructng dscrete D-optmal desgns, Computatonal statstcs and data analyss, pp (1992). [5] J. S. Jung and B. J. Yum, Constructon of exact D-optmal desgns by tabu search, Computatonal Statstcs and Data Analyss, pp (1996). [6] M. A. Lejeune, Heurstc optmzaton of expermental desgns, European Journal of Operatonal Research, pp (2003). [7] A. Björk, Numercal Methods for Least Squares Problems, ISBN (1996). [8] R. Ghanem and P. Spanos, Stochastc fnte elements: A Spectral Approach, Sprnger-Verlag ISBN or [9] G. Montepedra, Applcaton of genetc algorthms to the constructon of exact D-optmal desgns, Journal of Appled Statstcs, pages (1998). [10] R. C. St. John and N. R. Draper D-optmalty for regresson desgns: a revew, Technometrcs, pages (1975). [11] D.E. Goldberg, Genetc Algorthms n Search, Optmsaton and Machne Learnng, Cambrdge MA: Addson-Wesley (1989). [12] L. M. Hanes, The applcaton of the annealng algorthm to the constructon of exact optmal desgns for lnear-regresson models,technometrcs, vol. 29, num. 4 (1987). [13] R. H. Myers, D. C. Montgomery and G. G. Vnng, Generalzed Lnear Models wth Applcatons n Engneerng and the Scences, New York: Wley (2002).

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