INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERINGInt. J. Numer. Meth. Biomed. Engng. 2011; 27:1660–1664Published online 16 September 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.1412COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING

LETTER TO THE EDITOR

High-dimensional model representation for structural reliabilityanalysis: Authors’ reply to comments by S. Rahman and H. Xu

Rajib Chowdhury, B. N. Rao∗,† and A. Meher Prasad

Structural Engineering Division, Department of Civil Engineering, Indian Institute of Technology Madras,Chennai 600 036, India

Received 24 May 2010; Revised 27 July 2010; Accepted 28 July 2010

KEY WORDS: HDMR; structural reliability; failure probability

The purpose of the present communication is to provide response to the comments made byRahman and Xu (RX) [1] in relation to our paper on high-dimensional model representation(HDMR) for structural reliability analysis [2]. It is our belief that this response will have therole of correctly informing the readers about the real origin of HDMR concepts adopted in themanuscript [2] and erroneous claims of RX on the development of decomposition methods basedon HDMR concepts without giving appropriate credit to the original development of HDMR.Below we provide response to the points raised in RX’s letter.

Nowhere in the manuscript [2], we claimed inventing HDMR as our original contribution tothe literature. Hence, the comment, ‘. . .convey the impression that they are presenting a newmethodology and original contribution to the literature. . .’ under section 1 of RX’s letter [1] isincorrect. In fact, at the beginning of the last paragraph of introduction section we clearly mentionedthat the manuscript explores the potential of a new class of computational methods, referred to asHDMR [3–10], for predicting reliability of structural/mechanical systems subject to random loads,material properties, and geometry. All the papers related to the original development of HDMRhave been appropriately cited in the manuscript [2]. Hence the comment ‘. . .therefore the originaldevelopment of the decomposition predates the timeline specified in the authors’ response’ undersection 2 of RX’s letter [1] is a misleading statement intended to snow the reader with erroneousarguments. We leave to the original developers of HDMR concepts, to judge the truth in RX’scomments [1] such as ‘. . .Hickernell (1996) developed a reproducing kernel Hilbert space version.This decomposition has been lately examined by Rabitz and Alis (1999) for HDMR . . .’.

The concept of HDMR was originally developed by Rabitz’s group [3–9] for multivariatefunction approximation. Later after 4 years, Sobol [10] suggested to use the mean value of thevariables as reference points for HDMR approximation. Much later, it was Xu and Rahman [11]who reinvented the wheel without giving appropriate credit to the original development of HDMRby coining new terms such as ‘Decomposition methods’, ‘Univariate approximation’, ‘Bivariateapproximation’, ‘S-variate approximation’, in place of HDMR, first-, second-, and S-order HDMRapproximation, respectively. Regarding the selection of the reference point as the mean point, againRX [1] are making erroneous arguments and misleading statements. The selection of referencepoints as mean point suggested by Sobol [10] was also reinvented by Xu and Rahman [11]without giving appropriate credit to Sobol [10]. In our manuscript [2], citing [10] reference westated that ‘. . .it is shown that it is optimal to choose reference point c as mean values of the

∗Correspondence to: B. N. Rao, Structural Engineering Division, Department of Civil Engineering, Indian Instituteof Technology Madras, Chennai 600 036, India.

†E-mail: bnrao@iitm.ac.in

Copyright q 2010 John Wiley & Sons, Ltd.

HDMR FOR STRUCTURAL RELIABILITY ANALYSIS 1661

input variables. . .’. One can easily verify that in [10] under section 10, it was mentioned that‘As the reference point y, select the trial point x( j0) whose output value is nearest to f ∗

0 . . .’,

where f ∗0 =(1/N )

∑Nj=1 f (x( j)). Clearly, f ∗

0 designates the mean value. Hence, we leave to thereaders to judge whose response is replete with erroneous arguments and misleading statements,and whose has mistaken conclusion. In fact, even the proof that a truncated Cut-HDMR expansionis likely to give a better approximation of multivariate function than any truncated Taylor seriesbecause the latter only contains a finite number of terms has already been reported [6], which waslater reinvented again by Xu and Rahman [11] without giving appropriate credit. Hence, one caneasily verify that the ‘Decomposition methods’ were reinvented by Xu and Rahman [11] basedon HDMR concepts, and the same is also obvious from the comparison of striking similaritiesbetween ‘Decomposition methods’ and Cut-HDMR that RX themselves provided under section 2of their letter [1]. Surprisingly, one can verify that such a comparison of striking similarities hasnot been drawn in any of their papers (References [2, 7, 9] in RX’s letter) and no reference hasbeen given to any of the original development of HDMR [3–10]. Hence, we leave to the readersto judge erroneous claims of RX on multivariate function decomposition and the mean point asthe reference point, under section 2 of their letter [1]. While implementing the HDMR conceptsin the structural reliability analysis, we were not aware that there exist such striking similaritiesbetween ‘Decomposition methods’ and the original development of HDMR, as pointed out by RXunder section 2 of their letter [1]. Hence, in our paper [2] we cited only the papers related to theoriginal development of HDMR [3–10].

Prior to the study of reliability analysis using HDMR, we performed a through study [12]to explore potential of HDMR for tackling univariate and multivariate piece wise continuousfunctions. In that study [12], we showed adopting the same sampling scheme as that adoptedin [2], HDMR approximation using the Moving Least-Squares (MLS) interpolation scheme issuperior to that of the Lagrange interpolation, which motivated us to use it in our further studies[2]. Hence, in the current manuscript [2], we adopted the MLS interpolation scheme instead of theLagrange interpolation. In addition in the current manuscript [2], we demonstrated the potential ofHDMR, for predicting reliability of structural/mechanical systems having multi-dimensional non-differentiable limit state/performance function. Hence, the comment, ‘. . .only difference betweenthe Chowdhury et al. (2009) and Xu and Rahman (2005) papers is. . .’ under section 2 of RX’sletter [1] is incorrect.

In [13] where the response surface was utilized in conjunction with the advanced Monte CarloSimulation (MCS) techniques (importance sampling) to obtain the desired reliability estimates,one can easily verify that under section 3 it was mentioned (in their own words):

(a) The suggested way of obtaining g(x) is interpolation using points along the axes xi . Forthe following analysis the points xi are chosen to be the mean values xi , and xi = xi ± fi�i ,in which fi , an arbitrary factor ( fi =3 is employed in the numerical examples) and �i arethe standard deviations.

(b) Once g(x) is defined, the analysis can proceed in any suitable way, preferably using theadvanced MCS.

Sampling scheme similar to that suggested by [13] is adopted in our manuscripts [2, 12]. Inaddition, similar to that suggested by Bucher and Bourgund [13], in our manuscript [12] the failureprobability PF is estimated by performing MCS on first- and second-order approximation g(x) ofthe original implicit limit state/performance function g(x).

RX’s comment in their letter [1] ‘The generalized equation and/or its special cases do not existin the cut-HDMR method or any other HDMR-related work. . .’ is again erroneous and misleading,as the mth-order HDMR approximation has already been reported as Equation 9 in Rabitz andSim [14], which was later reinvented again by Xu and Rahman [15] without giving appropriatecredit to Rabitz and Sim [14]. In addition, one can easily verify that the decomposition methodsthat RX [1] are claiming to have developed based on the Taylor series expansion [15] are indeedbased on the proof that a truncated Cut-HDMR expansion is likely to give a better approximationof multivariate function than any truncated Taylor series because the latter only contains a finite

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. 2011; 27:1660–1664DOI: 10.1002/cnm

1662 R. CHOWDHURY, B. N. RAO AND A. M. PRASAD

number of terms, which has already been reported by Li et al. [16, 17], which was later reinventedagain by Xu and Rahman [15] without giving appropriate credit to Li et al. [16, 17].

Again RX’s comment in their letter [1] ‘. . .once we became cognizant of the HDMR methodsand realized the similarity between these two methods, we acknowledged the HDMR methods inour subsequent papers. . .’ is erroneous and misleading, as one can easily verify that in Rahmanand Wei [18], and Rahman [19] no acknowledgement was given to the HDMR methods. Even inRahman [20, 21], the HDMR methods were acknowledged possibly upon insistence of respectivemanuscripts’ reviewers.

All the terminology related to HDMR including first- and second-order HDMR approximationsare not coined by us, but were coined by Rabitz’s group [3–9] who were the original developersof the HDMR concepts. In addition, it is clearly mentioned in the manuscripts [2, 12] that HDMRexpresses the response g(x) as a hierarchical correlated function expansion in terms of the inputvariables as

g(x) = g0+N∑

i=1gi (xi )+ ∑

1�i1<i2�Ngi1i2(xi1, xi2)

+·· ·+ ∑

1�i1<···<il�Ngi1,i2,...,il (xi1, xi2, . . . , xil )+·· ·+g12...N (x1, x2, . . . , xN ), (1)

where g0 is a constant term representing the zeroth-order component function or the mean responseof g(x). The function gi (xi ) is a first-order term expressing the effect of variable xi acting alone,although generally nonlinearly, upon the output g(x). The function gi1i2(xi1, xi2) is a second-orderterm that describes the cooperative effects of the variables xi1 and xi2 upon the output g(x). Thehigher-order terms gives the cooperative effects of increasing numbers of input variables actingtogether to influence the output g(x). The last term g12,...,N (x1, x2, . . . , xN ) contains any residualdependence of all the input variables locked together in a cooperative way to influence the outputg(x). Considering terms up to first- and second-order in Equation (1) yields, respectively, first- andsecond-order approximations of g(x). Hence, RX’s comment in their letter [1] ‘. . .characterizingthese approximations by first- and second-order methods is confusing and possibly inaccuratebased on the traditional definition of the order of a function. . .’ shows their total disregard todistinct perspective in coining the terminology related to HDMR by the original developers of theHDMR concepts.

Regarding the reference for Example 7, we cited Penmetsa and Grandhi [22], as they were thefirst to study 10-bar truss example. As mentioned in the manuscript, the 10-bar truss structureis modeled using ADINA FE software. In order to limit the errors due to inherent differencesin the numerical tool adopted, we avoided comparison of the results obtained using the HDMRapproximation with FORM, SORM, and direct MCS results obtained using different FE analysissoftware reported in the existing literature. To be consistent with the results obtained using theHDMR approximation, for comparison purposes the failure probability is estimated with FORM,SORM, and direct MCS using the ADINA FE software and the limit state/performance functionwith critical threshold of displacement as 18 in, instead of 1.8 in as stipulated in Penmetsa andGrandhi [22]. As we found that the HDMR approximation using the MLS interpolation scheme issuperior to that of the Lagrange interpolation [12], and since the results presented in Table IX areself sufficient in all respects (to demonstrate the performance of HDMR approximation), resultsobtained using different FE analysis software and/or different interpolation schemes reported inthe existing literature are not included. Hence, the comment, ‘. . .Chowdhury et al. (2009) conveya misleading impression that they have developed a new reliability method and are presenting anoriginal contribution to the literature’ under section 3 of RX’s letter [1] is incorrect and exclusionof RX’s work from our reference list was not intentional. However, while preparing this letter, wecarefully read through the paper by Xu and Rahman [11]. In spite of the fact that the ‘Decompositionmethods’ were reinvented by Xu and Rahman [11] based on the HDMR concepts, we are sincere insaying that their work deserves to be cited and we are guilty of not citing the same. However, dueto lack of awareness on striking similarities between ‘Decomposition methods’ and the originaldevelopment of HDMR (pointed out by RX under section 2 of their letter [1]), we did not cite in

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. 2011; 27:1660–1664DOI: 10.1002/cnm

HDMR FOR STRUCTURAL RELIABILITY ANALYSIS 1663

our paper. Again we reiterate that all the papers related to the real origin of Example 9 have beenappropriately cited in the manuscript [2], and since the results presented in Table XIV are selfsufficient in all respects (to demonstrate the performance of HDMR approximation), the resultsobtained using different FE analysis software and/or different interpolation schemes reported inthe existing literature are not included.

We do agree on similarities (1) in the textual content of the introduction, which is basicallyliterature review summarizing the existing methods and previous researchers contributions; (2) inthe textual content of the problem description statement of Example 7; and (3) in the linguistic styleof the summary, introduction, and numerical examples sections. Such similarities in the textualcontent and linguistic style can even be noticed if one compares section 2.4 p. 241 of Xu andRahman [11], section 2.4 p. 2824 of Rahman and Wei [23], section 3 p. 2095 of Rahman [20],section 3 p. 1320 of Rahman [24], section 2 p. 2092 of Rahman [25] with section 1 p. 4 of Liet al. [16] and section 2.1 p. 7767 of Li et al. [17]. In addition, similarities in the linguistic stylebetween Chowdhury et al. [2] and Xu and Rahman [11] can be attributed to fact that the secondauthor (B. N. Rao) came from the same school of thought as Xu and Rahman, as B. N. Raodid his graduate studies at the University of Iowa and co-authored several papers with Rahmanin the past.

Finally, originality, accuracy, and completeness of the work presented in our manuscript [2] issummarized below:

Originality

We reiterate that we followed the original work of HDMR and used the same terminologycoined by the original developers. In our manuscript [2], we provided an exhaustive referencelist [3–10] from where we adopted the HDMR concepts. As mentioned earlier, in the manuscript[2] we adopted the MLS interpolation scheme instead of the Lagrange interpolation. In addi-tion, in the manuscript [2] we demonstrated the potential of HDMR, for predicting reliability ofstructural/mechanical systems having multi-dimensional non-differentiable limit state/performancefunction.

Accuracy

All the terminologies related to HDMR including first- and second-order HDMR approximationsare not coined by us, but were coined by Rabitz’s group who were the original developers of HDMRconcepts. Since considering terms up to first- and second-order in Equation 1 yields, respectively,first- and second-order approximation of g(x), they are termed as first- and second-order HDMRapproximations, respectively.

Completeness of work

As mentioned earlier, since the results presented in Tables IX and XIV are self sufficient in allrespects (to demonstrate the performance of HDMR approximation), the results obtained usingdifferent FE analysis software and/or different interpolation schemes reported in the existingliterature are not included.

In spite of the fundamental similarities in technical aspects between ‘Decomposition methods’and HDMR, surprisingly one could not find any reference to the original development of HDMReither in Xu and Rahman [11] or in references [13, 16] of RX’s letter. Hence, in the conclusion,it can be stated that it was RX who reinvented the wheel, coined new terminology, and claimedthat they presented an original contribution to the literature, which we leave to the readers tojudge. In spite of this fact, we are sincere in saying that their work deserves to be cited andwe are guilty of not citing the same. However, due to lack of awareness on striking similaritiesbetween ‘Decomposition methods’ and HDMR (pointed out by RX under section 2 of their letter[1]), we did not cite in our paper and exclusion of RX’s work from our reference list was notintentional. We believe that our explanation above provided response to all the points raised in RX’sletter [1].

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. 2011; 27:1660–1664DOI: 10.1002/cnm

1664 R. CHOWDHURY, B. N. RAO AND A. M. PRASAD

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Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. 2011; 27:1660–1664DOI: 10.1002/cnm