Statistical Moments of Polynomial Dimensional Decomposition

Transcription

1 Statitical Moment of Polynomial Dimenional Decompoition Sharif Rahman, M.ASCE 1 Abtract: Thi technical note preent explicit formula for calculating the repone moment of tochatic ytem by polynomial dimenional decompoition entailing independent random input with arbitrary probability meaure. The numerical reult indicate that the formula provide accurate, convergent, and computationally efficient etimate of the econd-moment propertie. DOI: / ASCE EM CE Databae ubject heading: Stochatic procee; Polynomial; Decompoition; Statitic. Author keyword: Stochatic mechanic; Orthogonal polynomial; Analyi of variance; High-dimenional model repreentation. Introduction Dimenional decompoition plit a multivariate function into a finite um of impler component function of input variable with increaing dimenion. The decompoition, firt preented by Hoeffding 1948 in relation to hi eminal work on U-tatitic, ha been tudied by many other reearcher, including Sobol 1969 for analyi of variance, Efron and Stein 1981 for jackknife etimate of variance, Rabitz and Ali 1999 for high-dimenional model repreentation, and Xu and Rahman 2005 for reliability analyi. More recently, the writer developed the polynomial dimenional decompoition PDD method involving Fourierpolynomial expanion of component function Rahman 2008, later extended for arbitrary probability meaure Rahman 2009, for tochatic computing. Thi tudy further examine the writer PDD method for calculating the repone moment of a complex tochatic ytem. By exploiting the orthogonal tructure of the decompoition and the propertie of orthogonal polynomial, explicit formula for calculating the repone moment in term of the expanion coefficient have been derived. The reult of an indutrial-cale leverarm example indicate that the formula provide accurate, convergent, and computationally efficient etimate of the firt two moment examined. ytem. The joint probability denity function of X i f X x = i=n i=1 f i x i, where f i x i =marginal probability denity function of X i defined on the probability triple i,f i, P i. Let y X, a realvalued, quare-integrable, meaurable tranformation on,f, define a relevant repone of the tochatic ytem. The PDD of y X, decribed by Rahman 2009 can be viewed a a finite, hierarchical expanion of an output function in term of it input variable with increaing dimenion, where ij X i, j=0,1,... i a et of complete orthonormal bae in the Hilbert pace L 2 i,f i, P i, which i conitent with the probability meaure of X i, and y 0 and C i1 i j 1 j, =1,2,...,N, are the expanion coefficient. In many application, the function y in Eq. 1 can be approximated by a um of at mot S-variate component function compriing at mot m-order orthogonal polynomial, where 1 S N and 1 m are both integer, reulting in the S-variate approximation Rahman Polynomial Dimenional Decompoition Let,F, P be a complete probability pace, where =ample pace; F= -field on ; and P:F 0,1 =probability meaure. With B N repreenting the Borel -field on R N, conider an R N -valued independent random vector X= X 1,...,X N T :,F R N,B N, which decribe input to a complex tochatic 1 Profeor, College of Engineering, The Univ. of Iowa, Iowa City, IA rahman@engineering.uiowa.edu Note. Thi manucript wa ubmitted on June 5, 2009; approved on November 9, 2009; publihed online on November 11, Dicuion period open until December 1, 2010; eparate dicuion mut be ubmitted for individual paper. Thi technical note i part of the Journal of Engineering Mechanic, Vol. 136, No. 7, July 1, ASCE, ISSN /2010/ /$ which converge to y X in the mean-quare ene when S=N and m. By minimizing an error functional aociated with a given y x and the joint probability denity of X i1,...,x i T, the coefficient can be expreed by the N-dimenional integral Rahman 2008 and y 0 = R N y x f X x dx 3 JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2010 / 923

2 C i1 i j 1 j y x ip j p x ip f X x dx, = R N =1,2,...,S 4 Once the embedded coefficient y 0 and C i1 i j 1 j, =1,2,...,S, are calculated, a decribed in the writer previou work Rahman 2009, Eq. 2 furnihe an approximate but explicit map ỹ S :R N R that can be viewed a a urrogate of the exact map y:r N R, which decribe the input-output relation from a complex numerical imulation. Therefore, any probabilitic characteritic of y X, including it tatitical moment and rare event probabilitie, can be eaily etimated by performing Monte Carlo imulation of ỹ S X rather than of y X. However, due to the pecial propertie of orthogonal polynomial, explicit formula for moment of ỹ S X can be derived, and that i the principal focu of thi work. Reader intereted in tail ditribution of repone or reliability analyi are referred to prior work on PDD Rahman 2008, Orthogonal Polynomial Let f i x i be the probability denity function of the random variable X i under the probability meaure df i x i = f i x i dx i, which ha finite moment of order up to 2m, m N. Let P be the pace of real polynomial and P m P the pace of polynomial of degree m. For any pair u i x i, v i x i in P, define an inner product u i,v i dfi u i x i v i x i df i x i u i x i v i x i f i x i dx i 5 ª R = R and an aociated norm u i dfi ª u i,u i dfi with repect to the meaure df i x i. Definition: Monic real polynomial ij x i =x j i +, j =0,1,2,..., are called monic orthogonal polynomial with repect to the meaure df i x i if ij1, ij2 dfi =0 for j 1 j 2, j 1, j 2 = 0,1,2,... and ij dfi = ij, ij dfi 0 for j = 0,1,2,... 6 There are infinitely many orthogonal polynomial if the index et j=0,1,2,... i unbounded and finitely many otherwie. Theorem: Let ij x i, j=0,1,2,..., be monic orthogonal polynomial with repect to the meaure df i x i. They atify the three-term recurrence relation i,j+1 x i = x i a ij ij x i b ij i,j 1 x i, j = 0,1,2,... poitive definite on P or P m, repectively, but not on P m, m m. Proof. See Gautchi 2004, pp The three-term recurrence relation ha ignificant impact on the contruction of orthogonal polynomial. The firt m recurion coefficient pair are uniquely determined by the firt 2m moment of X i that mut exit. When thee moment can be exactly calculated, they lead to exact recurion coefficient, ome of which belong to claical orthogonal polynomial. For an arbitrary meaure, approximate method baed on the Stieltje procedure can be employed to obtain the recurion coefficient. See Rahman 2009 for further detail. Let ij x i ª ij x i / ij x i dfi, j=0,1,2,..., define orthonormal verion of monic orthogonal polynomial for an arbitrary meaure df i.ife i the expectation operator, then two important propertie of ij x i are a follow. Property 1. The orthonormal polynomial bai function have a unit mean for j=0 and zero mean for all j 1, i.e. E ij X i ij x i f i x i dx i = ª R 1 if j = if j 1 Property 2. Any two orthonormal polynomial bai function ij1 X i and ij2 X i, where j 1, j 2 =0,1,2,..., are uncorrelated and each ha unit variance, i.e. E ij1 X i ij2 X i ij1 x i ij2 x i f i x i dx i = ª R 1 if j 1 = j 2 0 if j 1 j 2 11 The firt property tem from the expreion E ij X i = R ij x i f i x i dx i = ij, i0 dfi, which, according to the aforementioned definition and theorem, i one when j = 0 and zero when j 1. The econd property follow directly from the definition and theorem. Moment Applying the expectation operator on Eq. 2 and noting Property 1, the mean E ỹ S X = y 0 12 of the S-variate approximation matche the exact mean in Eq. 3, regardle of S. Applying the expectation operator again, thi time on ỹ S X y 0 2, reult in the approximate variance where i, 1 x i =0, i0 x i =1 7 a ij = x i ij, ij dfi ij, ij dfi, j = 0,1,2,... 8 and b ij = i0, i0 dfi if j =0 ij, ij dfi i,j 1, i,j 1 dfi if j =1,2,... 9 are the recurion coefficient. The index range i infinite j or finite j m 1, depending on whether the inner product i / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2010

3 which depend on S. The number of ummation inide the brace of the right ide of Eq. 13 i 2 +t, where and t=indice of the two outer ummation. By virtue of Property 2 and independent coordinate of X t 2 E ip j p X ip kp l p X kp = E ip j p X ip =1 14 for =t, i p =k p, j p =l p and zero otherwie, leading to Two lever arm Pin E Pin F Pin G (a) 15 a the um of quare of the expanion coefficient from the S-variate approximation of y x. Clearly, the approximate variance in Eq. 15 approache the exact variance P V 0.3 m Pin F u y =u yf, u z =0 y u y =u yg, u z =0 x Pin G P H 16 when S=N and m. The mean-quare convergence of ỹ S i guaranteed a y and it component function are all member of the aociated Hilbert pace. Therefore, Eq. 12 and 15 provide ueful formula for calculating the approximate mean and variance of a general tochatic repone. Compared with the pat work Rahman 2009, no imulation of ỹ S X i required to etimate the econd-moment tatitic of y X. Can the ame idea be extended to explore higher-order moment? For intance, applying the expectation operator on ỹ S X y 0 r, where r poitive integer, yield the rth-order central moment Pin E 1.36 m 1.72 m (b) u x =u xf, u z =0 u x =u xg, u z =0 (c) Fig. 1. Structural analyi of a leverarm: a two leverarm in a wheel loader; b geometry, loading, and boundary condition; and c undeformed meh 48,312 element For tatitically dependent random variable, a direct approach to PDD require contructing multivariate orthogonal polynomial for a general, multivariate joint denity function. New method avoiding nonlinear tranformation will need to be developed for generating meaure-conitent multivariate polynomial. Stochatic problem entailing dependent random variable are outide the cope of the preent work. 17 However, for r 2, expreing the expectation on the right ide of Eq. 17 in term of the expanion coefficient i hardly imple. Furthermore, expectation of product containing more than two orthonormal random polynomial are required. It i vital to emphaize that the PDD, like the polynomial chao expanion Field and Grigoriu 2004, doe not guarantee convergence of moment of order greater than two. Computational Expene The S-variate approximation in Eq. 2 require evaluation of the coefficient y 0 and C i1 i j 1 j, =1,...,S. If the coefficient are etimated by dimenion-reduction integration Rahman 2009 involving at mot R-dimenional S R N integration with an n-point quadrature rule, a employed here, the following determinitic repone function evaluation are required: y c, j y c 1,...,c 1 k1 1,x j k1,c R k1 +1,...,c kr 1,x kr,c kr +1,...,c N for k 1,..., k R =1,...,N and j 1,...,j R =1,...,n, where c= c 1,...,c N T JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2010 / 925

4 Table 1. Statitical Propertie of Leverarm Random Input Random variable Mean Standard deviation Probability ditribution P a H,kN Lognormal P a V,kN 1, Lognormal E, GPa Lognormal Lognormal u xf,mm 5 5/ 3 Uniform b u yf,mm 5 5/ 3 Uniform c u xg,mm 5 5/ 3 Uniform c u yg,mm 5 5/ 3 Uniform b a To be ditributed equally halved on front and back ide of pin E. b Uniformly ditributed over 10, 0 mm; to be applied on both ide. c Uniformly ditributed over 0, 10 mm; to be applied on both ide. =E X i the mean input and the upercript on the variable indicate correponding integration point. A a reult, the total cot for the S-variate PDD entail a maximum of k=0 k=r N R k n R k function evaluation. For intance, the univariate PDD S=R=1 and bivariate S=R=2 PDD will require only nn+1 linear in N or n and N N 1 n 2 /2+nN+1 quadratic in N or n function evaluation, repectively. Therefore, the PDD method employing dimenion-reduction integration hould be more efficient than crude Monte Carlo imulation for olving problem involving moderate number ay, le than a hundred of random variable. For higher-dimenional problem with hundred or thouand of random variable, the decompoition in Eq. 1 i till ueful, but more efficient method are needed to calculate the expanion coefficient. Regardle of the problem ize, the cot of generating meaure-conitent orthogonal polynomial and aociated Gau quadrature formula in PDD are negligible when compared with the calculation of the expanion coefficient. Numerical Example Conider a leverarm in a wheel loader depicted in Fig. 1 a, which i commonly ued in the heavy contruction indutry. The loading and boundary condition of a ingle leverarm are hown in Fig. 1 b. An undeformed leverarm meh from ABAQUS Simulia, Inc. 2008, which comprie 48,312 tetrahedral element, i preented in Fig. 1 c. Two random load P H and P V acting at pin E can be viewed a input load due to other mechanical component of the wheel loader. The eential boundary condition, ketched in Fig. 1 b, define random precribed diplacement u xf and u yf at pin F and u xg, and u yg at pin G. The leverarm i made of cat teel with random Young modulu E and random Poion ratio. The input vector X= P H, P V,E,,u xf,u yf,u xg,u yg T R 8 include eight independent random variable with the tatitical propertie lited in Table 1. Both univariate S=1 and bivariate S=2 PDD method with meaure-conitent orthogonal polynomial Rahman 2009 were employed to obtain the econdmoment tatitic of three elatic repone generated by linearelatic finite-element analyi FEA of the leverarm. The expanion coefficient were etimated by dimenion-reduction integration with R=S, where R=reduced dimenion, requiring oneor at mot two-dimenional integration Rahman The order m of orthogonal polynomial and number n of integration point in the dimenion-reduction integration are 1 m 3 and n=m+1, repectively. Table 2 preent the approximate mean and tandard deviation of maximum von Mie tre e,max, maximum larget principal train 1,max, and maximum ditortional energy denity U d,max of the entire leverarm by the univariate and bivariate PDD method. Thee elatic repone are commonly ued for examining material yielding or fatigue damage in mechanical ytem. The econd-moment tatitic by the PDD method in Table 2, calculated uing Eq. 12 and 15, quickly converge with repect to S and/or m. Compared with the pat work Rahman 2009, no imulation of ỹ S X wa needed or performed to gener- Table 2. Second-Moment Propertie of Leverarm Elatic Repone by Variou Method PDD Univariate S=1 Bivariate S=2 Crude Monte Carlo c Standard Number Standard Number Standard m Mean deviation of FEA a Mean deviation of FEA b Mean deviation Maximum von Mie tre e,max MPa Maximum larget principal train 1,max percent Maximum ditortional energy denity U d,max MPa a nn+1, where N=8, n=m+1 Rahman b N N 1 n 2 /2+nN+1, where N=8, n=m+1 Rahman c Sample ize=1, / JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2010

5 ate thee tatitic. Since FEA i employed for repone evaluation, the computational effort of PDD come primarily from numerically determining the expanion coefficient. The expene involved in etimating the PDD coefficient vary from FEA for the univariate PDD and FEA for the bivariate PDD, depending on the value of m. Since no exact olution exit, crude Monte Carlo imulation wa performed up to 1,000 realization. Compared with the Monte Carlo-generated tatitic, alo lited in Table 2, both verion of the PDD method provide excellent etimate of mean and tandard deviation of all three repone at lower computational cot. The univariate olution i not only accurate, but alo highly efficient. Thi i becaue of a realitic example choen, where the individual effect of input variable on the econd-moment tatitic of repone are dominant over their cooperative effect. However, for higher-order moment or tail probabilitie, not examined here, the univariate PDD may not be adequate; bivariate PDD entailing higher-order orthogonal polynomial may be required Rahman Concluion The PDD involving Fourier-polynomial expanion of lowerdimenional component function wa tudied. By exploiting the orthogonal tructure of the decompoition and the propertie of orthogonal polynomial, explicit formula for calculating the repone moment in term of the expanion coefficient were derived. The reult of an indutrial-cale leverarm problem indicate that the formula provide accurate and convergent etimate of the firt two moment examined at modet computational effort. Acknowledgment The writer acknowledge financial upport from the U.S. National Science Foundation under Grant No. CMMI Reference Efron, B., and Stein, C The jackknife etimate of variance. Ann. Stat., 9 3, Field, R. V., and Grigoriu, M. D On the accuracy of the polynomial chao approximation. Probab. Eng. Mech., , Gautchi, W Orthogonal polynomial: Computation and approximation, Oxford Univerity Pre, Oxford, U.K. Hoeffding, W A cla of tatitic with aymptotically normal ditribution. Ann. Math. Stat., 19 3, Rabitz, H., and Ali, O General foundation of high dimenional model repreentation. J. Math. Chem., , Rahman, S A polynomial dimenional decompoition for tochatic computing. Int. J. Numer. Method Eng., 76 13, Rahman, S Extended polynomial dimenional decompoition for arbitrary probability ditribution. J. Eng. Mech., , Simulia, Inc ABAQUS/tandard uer manual, verion 6.8, Simulia, Inc., Providence, R.I. Sobol, I. M Multidimenional quadrature formula and Haar function. Nauka, Mocow, 288 in Ruian. Xu, H., and Rahman, S Decompoition method for tructural reliability analyi. Probab. Eng. Mech., 20 3, JOURNAL OF ENGINEERING MECHANICS ASCE / JULY 2010 / 927