High dimensional model representation for piece-wise continuous function approximation

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2008; 24:1587–1609Published online 19 September 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.1053

High dimensional model representation for piece-wise continuousfunction approximation

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

SUMMARY

High dimensional model representation (HDMR) approximates multivariate functions in such a waythat the component functions of the approximation are ordered starting from a constant and graduallyapproaching to multivariance as we proceed along the terms like first-order, second-order and so on. Untilnow HDMR applications include construction of a computational model directly from laboratory/field data,creating an efficient fully equivalent operational model to replace an existing time-consuming mathematicalmodel, identification of key model variables, global uncertainty assessments, efficient quantitative riskassessments, etc. In this paper, the potential of HDMR for tackling univariate and multivariate piece-wisecontinuous functions is explored. Eight numerical examples are presented to illustrate the performance ofHDMR for approximating a univariate or a multivariate piece-wise continuous function with an equivalentcontinuous function. Copyright q 2007 John Wiley & Sons, Ltd.

Received 29 March 2007; Revised 27 July 2007; Accepted 14 August 2007

KEY WORDS: high dimensional model representation; univariate; multivariate; piece-wise continuous;function approximation

1. INTRODUCTION

High dimensional model representation (HDMR) [1–9] is a tool developed in order to expressinput–output relationships of complex, computationally burdensome models in terms of hierarchicalcorrelated function expansions. Application of HDMR methodology to even a complex nonlinearmodel provides an efficient means of obtaining an accurate reduced model of the original system.The uncertainty analysis of the outputs of the computationally burdensome model can then be well

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

†E-mail: [email protected]

Contract/grant sponsor: Board of Research in Nuclear Sciences, India; contract/grant number: 2004/36/39-BRNS/2332

Copyright q 2007 John Wiley & Sons, Ltd.

1588 R. CHOWDHURY, B. N. RAO AND A. M. PRASAD

approximated by a Monte Carlo analysis of the corresponding reduced model outputs, which isthus performed at a much lower computational cost without compromising accuracy. Until nowHDMR applications include construction of a computational model directly from laboratory/fielddata [1–4], creating an efficient fully equivalent operational model to replace an existing time-consuming mathematical model [5, 6], identification of key model variables [2], global uncertaintyassessments [7], efficient quantitative risk assessments [2], etc.

In this paper, the potential of HDMR for tackling univariate and multivariate piece-wisecontinuous functions is explored. The paper is organized as follows. Section 2 presents a briefoverview of HDMR. Section 3 presents response surface generation using HDMR. In Section 4presents principles of the moving least-squares (MLS) approximation for the construction of theshape/interpolation functions. Section 5 presents eight numerical examples to illustrate the appli-cation of HDMR to approximate a univariate or a multivariate piece-wise continuous function withan equivalent continuous function.

2. HIGH DIMENSIONAL MODEL REPRESENTATION

In recent years, there have been efforts to develop efficient methods to approximate multivariatefunctions in such a way that the component functions of the approximation are ordered starting froma constant and gradually approaching to multivariance as we proceed along the terms like first-order,second-order and so on. One such method is HDMR [1–9]. HDMR is a general set of quantitativemodel assessment and analysis tools for capturing the high dimensional relationships between setsof input and output model variables. Let the N -dimensional vector x={x1, x2, . . . , xN } with Nranging up to ∼102–103 or more represent the input variables of the model under consideration,and the output variable as f (x). Since the influence of the input variables on the output variable canbe independent and/or cooperative, HDMR expresses the output f (x) as a hierarchical correlatedfunction expansion in terms of the input variables as

f (x) = f0+N∑i=1

fi (xi )+ ∑1�i� j�N

fi j (xi , x j )+ ∑1�i� j�k�N

fi jk(xi , x j , xk)

+·· ·+ f12···N (x1, x2, . . . , xN ) (1)

where f0 denotes the mean response to f (x) which is a constant. The function fi (xi ) is a first-orderterm expressing the effect of variable xi acting alone, although generally nonlinearly, upon theoutput f (x). The function fi j (xi , x j ) is a second-order term that describes the cooperative effectsof the variables xi and x j upon the output f (x). The higher order terms give the cooperative effectsof increasing numbers of input variables acting together to influence the output f (x). The last termf12···N (x1, x2, . . . , xN ) contains any residual dependence of all the input variables locked togetherin a cooperative way to influence the output f (x). Once all the relevant component functionsin Equation (1) are determined and suitably represented, the component functions constitute anHDMR, thereby replacing the original computationally expensive method of calculating f (x) bythe computationally efficient model. Usually the higher order terms in Equation (1) are negligible[2] such that HDMR with only low order correlations to second-order [4] amongst the inputvariables is typically adequate in describing the output behavior. This has been verified in a numberof computational studies [7], where HDMR expansions up to second-order are often sufficient to

Copyright q 2007 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2008; 24:1587–1609DOI: 10.1002/cnm

HDMR FOR PIECE-WISE CONTINUOUS FUNCTION APPROXIMATION 1589

describe the outputs of many realistic systems. Therefore, it is expected that HDMR expansionconverges very rapidly.

Depending on the method adopted to determine the component functions in Equation (1) thereare two particular HDMR expansions: ANOVA-HDMR and cut-HDMR. ANOVA-HDMR is usefulfor measuring the contributions of the variance of individual component functions to the overallvariance of the output. On the other hand, cut-HDMR expansion is an exact representation of theoutput f (x) in the hyperplane passing through a reference point in the variable space.

In this work, cut-HDMR procedure is used for determining the component functions in approxi-mating a univariate or a multivariate piece-wise continuous function with an equivalent continuousfunction. Using the cut-HDMR method, first a reference point c={c1,c2, . . . ,cN } is defined in thevariable space. In the convergence limit, cut-HDMR is invariant to the choice of reference pointc. In practice, c is chosen within the neighborhood of interest in the input space. The expansionfunctions are determined by evaluating the input–output responses of the system relative to thedefined reference point c along associated lines, planes, subvolumes, etc. (i.e. cuts) in the inputvariable space. This process reduces to the following relationship for the component functions inEquation (1):

f0 = f (c) (2)

fi (xi ) = f (xi ,ci )− f0 (3)

fi j (xi , x j ) = f (xi , x j ,ci j )− fi (xi )− f j (x j )− f0 (4)

where the notation f (xi ,ci )= f (c1,c2, . . . ,ci−1, xi ,ci+1, . . . ,cN ) denotes that all the input vari-ables are at their reference point values except xi . The f0 term is the output response of the systemevaluated at the reference point c. The higher order terms are evaluated as cuts in the input variablespace through the reference point. Therefore, each first-order term fi (xi ) is evaluated along itsvariable axis through the reference point. Each second-order term fi j (xi , x j ) is evaluated in a planedefined by the binary set of input variables xi , x j through the reference point; etc. The processof subtracting the lower order expansion functions removes their dependence to assure a uniquecontribution from the new expansion function.

3. RESPONSE SURFACE GENERATION

HDMR in Equation (1) is exact along any of the cuts, and the output response f (x) at a point xoff of the cuts can be obtained by following the procedures in steps 1 and 2.

Step 1: Interpolate each of the low-dimensional HDMR expansion terms with respect to theinput values of the point x. For example, consider the first-order component function f (xi ,ci )=f (c1,c2, . . . ,ci−1, xi ,ci+1, . . . ,cN ). If for xi = x j

i ,n function values

f (x ji ,ci )= f (c1, . . . ,ci−1, x

ji ,ci+1, . . . ,cN ), j=1,2, . . . ,n (5)

are given at n(=3,5,7 or 9) uniformly distributed sample points �i −(n−1)�i/2, �i −(n−3)�i/2, . . . ,�i , . . . ,�i +(n−3)�i/2,�i +(n−1)�i/2 along the variable axis xi with mean �i and



standard deviation �i , the function value for arbitrary xi can be obtained by the MLS [10]interpolation as

f (xi ,ci )=n∑j=1

� j (xi ) f (c1, . . . ,ci−1, xji ,ci+1, . . . ,cN ) (6)

Similarly, consider the second-order component function f (xi1, xi2,ci1i2)= f (c1, . . . ,ci1−1, xi1,

ci1+1, . . . ,ci2−1, xi2,ci2+1, . . . ,cN ). If for xi1 = x j1i1

and xi2 = x j2i2

,n2 function values

f (x j1i1

, x j2i2

,ci1i2)= f (c1, . . . ,ci1−1, xj1i1

,ci1+1, . . . ,ci2−1, xj2i2

,ci2+1, . . . ,cN )

j1=1,2, . . . ,n, j2=1,2, . . . ,n (7)

are given on a grid formed by taking n(=3,5,7 or 9) uniformly distributed sample points�i1 −(n−1)�i1/2,�i1 −(n−3)�i1/2, . . . ,�i1, . . . ,�i1 +(n−3)�i1/2,�i1 +(n−1)�i1/2 along xi1axis with mean �i1 and standard deviation �i1 , and n(=3,5,7 or 9) uniformly distributed samplepoints �i2 −(n−1)�i2/2, �i2 −(n−3)�i2/2, . . . ,�i2, . . . ,�i2 +(n−3)�i2/2,�i2 +(n−1)�i2/2 alongxi2 axis with mean �i2 and standard deviation �i2 , the function value for arbitrary (xi1, xi2) can beobtained by the MLS interpolation [10] as

f (xi1, xi2,ci1i2) =

n∑j1=1

n∑j2=1

� j1 j2(xi1, xi2)

× f (c1, . . . ,ci1−1, xj1i1

,ci1+1, . . . ,ci2−1, xj2i2

,ci2+1, . . . ,cN ) (8)

where the interpolation functions � j (xi ) and � j1 j2(xi1, xi2) are obtained either by using theLagrange interpolation or by using the MLS interpolation scheme [10] explained in the nextsection.

By using Equation (6), fi (xi ) can be generated if n function values are given at correspondingsample points. Similarly, by using Equation (8), fi1i2(xi1, xi2) can be generated if n2 functionvalues at corresponding sample points are given. The same procedure shall be repeated for allthe first-order component functions, i.e. fi (xi ), i=1,2, . . . ,N , and the second-order componentfunctions, i.e. fi1i2(xi1, xi2); i1, i2=1,2, . . . ,N .

Step 2: Sum the interpolated values of HDMR expansion terms from zeroth order to the highestorder retained in keeping with the desired accuracy. This leads to first-order approximation of thefunction f (x) as

f (x)=N∑i=1

n∑j=1

� j (xi ) f (c1, . . . ,ci−1, xji ,ci+1, . . . ,cN )−(N−1) f0 (9)

and second-order approximation of the function f (x) as

f (x) =N∑

i1=1,i2=1i1<i2

n∑j1=1

n∑j2=1

� j1 j2(xi1, xi2) f (c1, . . . ,ci1−1, xj1i1

,ci1+1, . . . ,ci2−1, xj2i2

,ci2+1, . . . ,cN )

−(N−2)N∑i=1

n∑j=1

� j (xi ) f (c1, . . . ,ci−1, xji ,ci+1, . . . ,cN )+ (N−1)(N−2)

2f0 (10)



where � j1 j2(xi1, xi2) is the shape/interpolation function obtained either by using the Lagrangeinterpolation or by using the MLS interpolation scheme [10]. The shape/interpolation function� j (xi ) in Equation (6) using the Lagrange interpolation is defined as

� j (xi )=(xi −x (1)

i ) · · · (xi −x ( j−1)i )(xi −x ( j+1)

i ) · · · (xi −x (n)i )

(x ( j)i −x (1)

i ) · · · (x ( j)i −x ( j−1)

i )(x ( j)i −x ( j+1)

i ) · · ·(x ( j)i −x (n)

i )(11)

and the shape/interpolation function � j1 j2(xi1, xi2) in Equation (8) using the Lagrange interpolationis defined as

� j1 j2(xi1, xi2) =[

(xi1 −x (1)i1

) · · · (xi1 −x ( j1−1)i1

)(xi1 −x ( j1+1)i1

) · · ·(xi1 −x (n)i1

)

(x ( j1)i1

−x (1)i1

) · · · (x ( j1)i1

−x ( j−1)i1

)(x ( j1)i1

−x ( j+1)i1

) · · · (x ( j1)i1

−x (n)i1

)

]

×[

(xi2 −x (1)i2

) · · · (xi2 −x ( j2−1)i2

)(xi2 −x ( j2+1)i2

) · · · (xi2 −x (n)i2

)

(x ( j2)i2

−x (1)i2

) · · ·(x ( j2)i2

−x ( j2−1)i2

)(x ( j2)i2

−x ( j2+1)i2

) · · ·(x ( j2)i2

−x (n)i2

)

](12)

Computation of the shape/interpolation functions � j (xi ) in Equation (6) and � j1 j2(xi1, xi2) inEquation (8) using the MLS interpolation function is explained in the next section. In the presentstudy it is found that the approximation using the MLS interpolation scheme is superior to thatof the Lagrange interpolation, which is demonstrated in Example 7. A flow diagram for responsesurface generation using HDMR is shown in Figure 1.

If n is the number of sample points taken along each of the variable axis and s is the order ofthe component function considered, starting from zeroth order to lth order, then the total numberof function evaluation for interpolation purpose is given by

l∑s=0

N !(N−1)!s! (n−1)s (13)

which grows polynomially with n and s. As a few low-order component functions of HDMRare used, the sample savings due to HDMR are significant compared with traditional sampling.Hence, uncertainty analysis using HDMR relies on an accurate reduced model being generatedwith a small number of full model simulations. An arbitrarily large sample Monte Carlo analysiscan be performed on the outputs approximated by HDMR which result in the same distributionsas obtained through the Monte Carlo analysis of the full model. The tremendous computationalsavings result from just having to perform interpolation instead of full model simulations for outputdetermination.

4. MOVING LEAST-SQUARES APPROXIMATION

Consider a function, u(x) over a domain, �⊆�K , where K =1,2 or 3. Let �x⊆� denote asub-domain describing the neighborhood of a point, x∈�K located in �. According to the movingleast squares [10], the approximation, uh(x) of u(x), is

uh(x)=m∑i=1

pi (x)ai (x)=pT(x)a(x) (14)



Figure 1. Flowchart of HDMR based response surface generation.

where pT(x)={p1(x), p2(x), . . . , pm(x)} is a vector of complete basis functions of order m anda(x)={a1(x),a2(x), . . . ,am(x)} is a vector of unknown parameters that depend on x . The basisfunctions should satisfy the following properties: (1) p1(x)=1; (2) pi (x)∈Cs(�), i=1,2, . . . ,m,where Cs(�) is a set of functions that have continuous derivatives up to order s on �; and (3)pi (x), i=1,2, . . . ,m, constitute a linearly independent set. For example, in one dimension (K =1)with x1-coordinate

pT(x)={1, x1}, m=2 (15)

and

pT(x)={1, x1, x21}, m=3 (16)



representing linear and quadratic basis functions, respectively. Similarly, in two dimensions (K =2)with x1- and x2-coordinates linear and quadratic basis functions are, respectively,

pT(x)={1, x1, x2}, m=3 (17)

and

pT(x)={1, x1, x2, x21 , x1x2, x22}, m=6 (18)

In Equation (14), the coefficient vector, a(x), is determined by minimizing a weighted discreteL2 norm, defined as

J (x)=nt∑I=1

wI (x)[pT(xI )a(x)−dI ]2=[Pa(x)−d]TW[Pa(x)−d] (19)

where xI denotes the coordinates of the sample point I , dT={d1,d2, . . . ,dnt } with dIrepresenting the nodal parameter (not the nodal values of uh(x)) for sample point I , W=diag[w1(x),w2(x), . . . ,wnt (x)] with wI (x) denoting the weight function associated with thesample point I such that wI (x)�0 for all x in the support �x of wI (x) and zero otherwise, nt isthe number of sample points in the domain �, and

P=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

pT(x1)

pT(x2)

...

pT(xnt )

⎤⎥⎥⎥⎥⎥⎥⎥⎦

∈L(�nt ×�m) (20)

A number of weight functions are available in the literature [11]. In this study, a weight functionproposed by Rao and Rahman [12] is used, which is defined as

wI (x)=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(1+�2

z2Iz2mI

)−((1+�)/2)

−(1+�2)−((1+�)/2)

1−(1+�2)−((1+�)/2), zI�zmI

0, zI>zmI

(21)

where � is a parameter controlling the shape of the weight function, zI =‖x−xI‖ distance froma point x to the sample point I , and zmI is the domain of influence of the sample point I . Thestationarity of J (x) with respect to a(x) yields

A(x)a(x)=C(x)d (22)

where

A(x) =nt∑I=1

wI (x)p(xI )pT(xI )=PTWP (23)

C(x) = [w1(x)p(x1), . . . ,wn(x)p(xn)]=PTW (24)



Solving a(x) from Equation (22) and then substituting it in Equation (14) yields

uh(x)=nt∑I=1

�I (x)dI =�T(x)d (25)

where

�T(x)={�1(x),�2(x), . . . ,�nt (x)}=pT(x)A−1(x)C(x) (26)

is a vector with its I th component

�I (x)=m∑j=1

p j (x)[A-1(x)C(x)] j I (27)

representing the interpolation function of the MLS approximation corresponding to the samplepoint I .

5. NUMERICAL EXAMPLES

An exact continuous function to replace a univariate or a multivariate piece-wise continuousfunction may not always be available in general problems. Rather than seeking an exact continuousfunction to replace a piece-wise continuous function, an equivalent continuous function can befound based on the HDMR concept. Eight numerical examples are presented to illustrate theperformance of HDMR to approximate a univariate or a multivariate piece-wise continuousfunction with an equivalent continuous function. For first-order approximation, n(=3,5,7 or 9)uniformly distributed sample points �i −(n−1)�i/2,�i −(n−3)�i/2, . . . ,�i , . . . ,�i +(n−3)�i/2,�i +(n−1)�i/2 are deployed along the variable axis xi with mean �i and standard deviation �i ,through the reference point. The sampling scheme for first-order HDMR for a function havingone variable (x) and two variables (x1 and x2) is shown in Figure 2(a) and (b), respectively.

x

c

x1

x2

(a) (b)

Figure 2. Sampling scheme for first-order HDMR: (a) for a function having one variable (x) and (b) fora function having two variables (x1 and x2).



x 1

x 2

c

Figure 3. Sampling scheme for second-order HDMR for a function having two variables (x1 and x2).

For second-order approximation, a regular grid is formed by taking n(=3,5,7 or 9) uniformlydistributed sample points �i1 −(n−1)�i1/2,�i1 −(n−3)�i1/2, . . . ,�i1, . . . ,�i1 +(n−3)�i1/2,�i1 +(n−1)�i1/2 along the xi1 axis with mean �i1 and standard deviation �i1 , and n(=3,5,7 or 9)uniformly distributed sample points �i2 −(n−1)�i2/2,�i2 −(n−3)�i2/2, . . . ,�i2, . . . ,�i2 +(n−3)�i2/2,�i2 +(n−1)�i2/2 along the xi2 axis with mean �i2 and standard deviation �i2 , throughthe reference point. The sampling scheme for second-order HDMR for a function having twovariables (x1 and x2) is shown in Figure 3. The total number of function evaluation involved in first-order and second-order approximations is (n−1)N+1 and (n−1)2(N−1)N/2+(n−1)N+1,respectively, in all examples. In the numerical examples, for first-order and second-order HDMRapproximation, the shape/interpolation functions � j (xi ) in Equation (6) and � j1 j2(xi1, xi2) inEquation (8) are obtained by using the MLS interpolation scheme. In Example 7, for second-orderHDMR approximation, both the MLS interpolation and the Lagrange interpolation schemesare adopted, to study the effect of the interpolation scheme on HDMR approximation. In allthe numerical examples, to demonstrate the performance of HDMR, the approximation errorgiven by

error=√(∫

�(F− F)2 d�

)/∫�F2 d�×100 (28)

is computed, where F is the actual function and F is the approximate equivalent continuousfunction.

Example 1In this example one-dimensional function of the following form is studied:

F=|x | (29)



-4 -2 0 2 4

x

0

1

2

3

4

F(x)

Exact

HDMR approximation

Figure 4. Original function and first-order HDMR approximation (Example 1).

The function is piece-wise continuous at x=0 and is linear in both the segments x<0 and x>0.For first-order HDMR approximation, five uniformly distributed sample points (n=5) in the range−2�x�2, with reference point c=0, are deployed along the x-axis, i.e. {x1, x2, x3, x4, x5}={−2,−1,0,1,2}. Figure 4 shows the effectiveness of first-order HDMR in approximating a piece-wise continuous function given in Equation (29) with an equivalent continuous function. Theapproximation error computed using Equation (28) is 0.95%.

Example 2Another one-dimensional function of the following form:

F(x)=⎧⎨⎩x2, x�0

x, x>0(30)

is considered. Here the function is piece-wise continuous at x=0 having a quadratic variationin the segment x<0 and a linear variation in the segment x>0. Similar to Example 1, fiveuniformly distributed sample points (n=5) in the range −2�x�2, with reference point c=0,are considered along the x-axis for first-order HDMR approximation, i.e. {x1, x2, x3, x4, x5}={−2,−1,0,1,2}. Figure 5(a) compares the original and the approximation of the function inEquation (30) using first-order HDMR with five sample points. The approximation error computedusing Equation (28) is 2.12%. In an effort to reduce the approximation error further, seven uniformlydistributed sample points n=7 in the range −2�x�2, with reference point c=0, are consideredalong the x-axis for first-order HDMR approximation. Figure 5(b) compares the original andthe approximation of the function using first-order HDMR with seven sample points. Increasein the number of sample points resulted in reduction of the approximation error dramaticallyto 0.16%.



-4 -2 0 2 4

x

0

1

2

3

4

5

6

7

8

F(x)

Exact

HDMR approximation (n = 5)

-4 -2 0 2 4x

0

1

2

3

4

5

6

7

8

F(x)

Exact


(a) (b)

Figure 5. Original function and first-order HDMR approximation (Example 2): (a) n=5 and (b) n=7.

-4 -2 0 2 4x

F(x )

Exact


-4 -2 0 2 4x

-1.00

-0.50

0.00

0.50

1.00

1.50

-1.00

-0.50

0.00

0.50

1.00

1.50F(

x)

Exact


(a) (b)

Figure 6. Original function and first-order HDMR approximation (Example 3): (a) n=5 and (b) n=7.

Example 3In this example the following piece-wise continuous function is studied:

F(x)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−sin(x), x<0

x2, 0�x�1

1

x, x>1

(31)

The function in Equation (31) is piece-wise continuous at x=0 and 1, with three different non-linear variations in the segments x<0, 0�x�1 and x>1. Similar to the previous two examples,first-order HDMR approximation with five and seven uniformly distributed sample points in therange −2�x�2, with reference point c=0, is studied. Figure 6(a) and (b) compares the originaland the approximation of the function in Equation (31) using first-order HDMR with n=5 and 7,respectively. The approximation error computed using Equation (28) for first-order HDMR with



n=5 and 7 is, respectively, 12.50 and 8.81%, which demonstrates reduction in the approximationerror with increase in the number of sample points.

Example 4Consider a rhombic function defined as

F(x, y)=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

x+ y−1, 0�x, y�1

−x+ y−1, −1�x�0, 0�y�1

−x− y−1, −1�x�0, −1�y�0

x− y−1, 0�x�1, −1�y�0

(32)

The function is piece-wise continuous at (x, y)=(0,0) having a bi-linear variation in differentregions of the domain. First-order HDMR approximation is studied by deploying five (n=5) andseven (n=7) uniformly distributed sample points along the x- and y-axes, and the reference pointis taken as c=(0,0) in the ranges −1�x�1 and −1�y�1. First-order HDMR approximation withfive and seven sample points involves 9 and 13 function evaluations, respectively. Figure 7 displaysthe plots and contours of the original function in Equation (32). Figure 8(a) and (b) displays,respectively, the plots and contours of the approximate function by first-order HDMR obtainedusing n=5 and 7, respectively. The approximation error is, respectively, 0.85 and 0.27% for first-order HDMR obtained using n=5 and 7. In an effort to reduce the approximation error further,second-order HDMR approximation is used with five and seven uniformly distributed sample pointsalong the x- and y-axes to form a regular grid. Figure 9(a) and (b) displays, respectively, the plotsand contours of the approximate function by second-order HDMR obtained using n=5 and 7,respectively. Second-order HDMR approximation resulted in reduction of the approximation errorto 0.13 and 0.06%, respectively, for n=5 and 7. However, the total number of function evaluationsrequired for second-order HDMR approximation using n=5 and 7 is, respectively, 25 and 49,which is comparatively more than for first-order HDMR approximation.

-0.8

-0.8-0.6-0.6

-0.6-0.4

-0.4

-0.4

-0.4

-0.4

-0.2

-0.2

-0.2

-0.2

-0.2-0.2

-0.2

0

00

0

0

0

0

0

0.2

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0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

-1-0.5

00.5

1

-1-0.5

00.5

1-1

-0.5

0

0.5

1

F (x

, y)

x y

(a) (b)

Figure 7. Rhombic function in Equation (32) (Example 4): (a) plot and (b) contour.



x

y

-1-0.5

00.5

1

-1-0.5

00.5

1-1

-0.5

0

0.5

1

-0.8

-0.8

-0.6-0.6

-0.6

-0.4

-0.4

-0.4

-0.4

-0.4

-0.2

-0.2 -0.2

-0.2

-0.2

-0.2

-0.2

0

00

0

0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

F (x

, y)

x

-1-0.5

00.5

1

-1-0.5

00.5

1-1

-0.5

0

0.5

1

-0.8

-0.8

-0.6

-0.6-0.6

-0.4

-0.4

-0.4

-0.4

-0.4

-0.2

-0.2 -0.2

-0.2

-0.2

-0.2

-0.2

0

0

0

0

0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

F (x

, y)

x

y

y

y

x

(a)

(b)

Figure 8. First-order HDMR approximation of a rhombic function in Equation (32) (Example 4):(a) plot and contour (n=5) and (b) plot and contour (n=7).

Example 5In this example a two-dimensional function of the following form:

F(x, y)=⎧⎨⎩x2+ y2, x, y�0

x+ y, x, y>0(33)

is considered. The function is piece-wise continuous at (x, y)=(0,0) having a quadratic variationin the segment x, y�0 and a bi-linear variation in the segment x, y>0. Similar to that in Example 4,first-order HDMR approximation is studied by deploying five (n=5) and seven (n=7) uniformlydistributed sample points along the x- and y-axes, and the reference point is taken as c=(0,0)in the ranges −2�x�2 and −2�y�2. Figure 10 displays the plots and contours of the originalfunction in Equation (33). Figure 11(a) and (b) displays, respectively, the plots and contoursof the approximate function by first-order HDMR obtained using n=5 and 7, respectively. Theapproximation error is, respectively, 3.91 and 2.06% for first-order HDMR obtained using n=5and 7. Again similar to that in Example 4, second-order HDMR approximation is studied to reduce



x

y

-1-0.5

00.5

1

-1-0.5

00.5

1-1

-0.5

0

0.5

1

-0.8

-0.8

-0.6-0.6

-0.6

-0.4

-0.4

-0.4

-0.4

-0.4

-0.2

-0.2 -0.2

-0.2

-0.2

-0.2

-0.2

0

00

0

0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

F (x

, y)

x

-1-0.5

00.5

1

-1-0.5

00.5

1-1

-0.5

0

0.5

1

-0.8

-0.8

-0.6

-0.6-0.6

-0.4

-0.4

-0.4

-0.4

-0.4

-0.2

-0.2 -0.2

-0.2

-0.2

-0.2

-0.2

0

0

0

0

0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

F (x

, y)

x

y

x

y

y

(a)

(b)

Figure 9. Second-order HDMR approximation of rhombic function in Equation (32) (Example 4):(a) plot and contour (n=5) and (b) plot and contour (n=7).

-2-1

01

2 -2-1

01

20

1

2

3

4

5

6

7

8

F (x

, y)

xy

x

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

4

5

5

6

6

7

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

y

(a) (b)

Figure 10. Function in Equation (33) (Example 5): (a) plot and (b) contour.



x

yy

1

1

1

2

2

2

3

3

3

3

4

4

4

4

4

5

56

6

7

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

x

6

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-2

-2

-1-10

0

1

1

2

20

1

2

3

4

5

6

7

8

y

-2

-2

-1-10

0

1

1

2

2

yx

x

F (x

, y)

F (x

, y)

0

1

2

3

4

5

6

7

8

1

1

1

2

2

2

3

3

3

3

4

4

4

4

4

5

5

6

7

(a)

(b)

Figure 11. First-order HDMR approximation of a function in Equation (33) (Example 5): (a) plot andcontour (n=5) and (b) plot and contour (n=7).

the approximation error, with five and seven uniformly distributed sample points along the x- andy-axes to form a regular grid. Figure 12(a) and (b) displays, respectively, the plots and contoursof the approximate function by second-order HDMR obtained using n=5 and 7, respectively.Second-order HDMR approximation resulted in reduction of the approximation error dramaticallyto 0.07 and 0.03%, respectively, for n=5 and 7.

Example 6In this example a function defining a portion of the intersection of the cylinders

x2+z2=1 and y2+z2=1 (34)

in the ranges −2�x�2 and −2�y�2 is considered. Figure 13 displays the plots and contours ofthe original function defining a portion of the intersection of the cylinders given in Equation (34).



(a)

(b)

y

F (x

, y)

F (x

, y)

x

x

y

-2-1 0

12 -2

-10

12

yx

-2-1 0

12 -2

-10

12

0

1

2

3

4

5

6

7

8

1

1

1

2

22

2

3

3

3

3

4

4

4

4

4

5

5

6

6

7

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

x-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

y

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0

1

2

3

4

5

6

7

8

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

56

7

4

Figure 12. Second-order HDMR approximation of a function in Equation (33) (Example 5): (a) plot andcontour (n=5) and (b) plot and contour (n=7).

-2-1

01

2

-2

-1

0

1

2-3

-2

-1

0

1

F (x

, y)

xy

x

y

-3-3

-3

-3

-3-3

- 2. 5

-2.5-2.5

-2.5-2.5

- 2. 5

-2-2

-2

-2-2

-2

-1.5-1.5

-1.5

-1.5

-1.5-1.5

-1-1

-1

-1

-1-1

-0.5-0.5

-0.5

- 0.5

-0.5-0.5

00

0

00

00.50.5

0.5

0.50.5

0.511

11

11

-3 -3 -3

-3 -3 -3

-2.5 -2.5 -2.5

-2.5 -2.5 -2.5

-2 -2 -2

-2 -2 -2

-1.5 -1.5 -1.5

-1.5 -1.5 -1.5

-1 -1 -1

-1 -1 -1

-0.5 -0.5 -0.5

-0.5 -0.5 -0.5

0000

0000

0.5 0.5 0.5

0.5 0.5 0.5

1111 1111

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(a) (b)

Figure 13. Function defining a portion of intersection of two cylinders in Equation (34)(Example 6): (a) plot and (b) contour.



(a)

-3

-2

-1

0

1

F (x

, y)

y

(b)

-2-1

01

2

-2-1

01

2-3

-2

-1

0

1

y

F (x

, y)

x

-2-1

01

2

-2-1

01

2

yx

y

x

33

3

-3

-3-3

- 2.5

- 2.5

-2.5

-2.5

-2.5

-2.5

-2-2

-2

-2-2

-2

-1.5-1.5

-1.5

-1.5

-1. 5

-1.5

-1-1

-1

-1-1

-1

-0.5

- 0.5

-0.5

-0.5-0.5

-0.500

0

00

00.50.5

0.5

0.50.5

0.5

11

11

11-3 -3 -3

-3 -3 -3

-2.5 -2.5 -2.5

-2.5 -2.5 -2.5

-2 -2 -2

-2 -2 -2

-1.5 -1.5 -1.5

-1.5 -1.5 -1.5

-1 -1 -1

-1 -1 -1

-0.5 -0.5 -0.5

-0.5 -0.5 -0.5

0000

0000

0.5 0.5 0.5

0.5 0.5 0.5

1111 1111

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

33

3

-3

-3-3

- 2.5

- 2.5

-2.5

-2.5

-2.5

-2.5

-2-2

-2

-2-2

-2

-1.5-1.5

-1.5

-1.5

-1. 5

-1.5

-1-1

-1

-1-1

-1

-0.5

- 0.5

-0.5

-0.5-0.5

-0.500

0

00

00.50.5

0.5

0.50.5

0.5

11

11

11-3 -3 -3

-3 -3 -3

-2.5 -2.5 -2.5

-2.5 -2.5 -2.5

-2 -2 -2

-2 -2 -2

-1.5 -1.5 -1.5

-1.5 -1.5 -1.5

-1 -1 -1

-1 -1 -1

-0.5 -0.5 -0.5

-0.5 -0.5 -0.5

0000

0000

0.5 0.5 0.5

0.5 0.5 0.5

1111 1111

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 14. First-order HDMR approximation of function defining a portion of intersection of two cylindersin Equation (34) (Example 6): (a) plot and contour (n=5) and (b) plot and contour (n=7).

The function is having a quadratic variation over the whole domain. Similar to that in Example 4,first-order HDMR approximation is studied by deploying five (n=5) and seven (n=7) uniformlydistributed sample points along the x- and y-axes, and the reference point is taken as c=(0,0)in the ranges −2�x�2 and −2�y�2. Figure 14(a) and (b) displays, respectively, the plots andcontours of the approximate function by first-order HDMR obtained using n=5 and 7, respectively.Again similar to that in Example 5, second-order HDMR approximation is studied, with fiveand seven uniformly distributed sample points along the x- and y-axes to form a regular grid.Figure 15(a) and (b) displays, respectively, the plots and contours of the approximate function bysecond-order HDMR obtained using n=5 and 7, respectively. The approximation error is almostzero for first-order and second-order HDMR approximations obtained using n=5 and 7, therebydemonstrating the effectiveness of HDMR in approximating the intersection of the two cylindersaccurately.



(a) -2-1

01

2

-2

-1

0

1

2-3

-2

-1

0

1

y

F (x

, y)

F (x

, y)

xx

y

33

3

-3

-3-3

- 2.5

- 2.5

-2.5

-2.5

-2.5

-2.5

-2-2

-2

-2-2

-2

-1.5-1.5

-1.5

-1.5

-1. 5

-1.5

-1-1

-1

-1-1

-1

-0.5

- 0.5

-0.5

-0.5-0.5

-0.500

0

00

00.50.5

0.5

0.50.5

0.5

11

11

11-3 -3 -3

-3 -3 -3

-2.5 -2.5 -2.5

-2.5 -2.5 -2.5

-2 -2 -2

-2 -2 -2

-1.5 -1.5 -1.5

-1.5 -1.5 -1.5

-1 -1 -1

-1 -1 -1

-0.5 -0.5 -0.5

-0.5 -0.5 -0.5

0000

0000

0.5 0.5 0.5

0.5 0.5 0.5

1111 1111

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(b)-2

-10

12

-2

-1

0

1

2-3

-2

-1

0

1

yx

x

y

33

3

-3

-3-3

- 2.5

- 2.5

-2.5

-2.5

-2.5

-2.5

-2-2

-2

-2-2

-2

-1.5-1.5

-1.5

-1.5

-1. 5

-1.5

-1-1

-1

-1-1

-1

-0.5

- 0.5

-0.5

-0.5-0.5

-0.500

0

00

00.50.5

0.5

0.50.5

0.5

11

11

11-3 -3 -3

-3 -3 -3

-2.5 -2.5 -2.5

-2.5 -2.5 -2.5

-2 -2 -2

-2 -2 -2

-1.5 -1.5 -1.5

-1.5 -1.5 -1.5

-1 -1 -1

-1 -1 -1

-0.5 -0.5 -0.5

-0.5 -0.5 -0.5

0000

0000

0.5 0.5 0.5

0.5 0.5 0.5

1111 1111

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 15. Second-order HDMR approximation of a function defining a portion of intersection of twocylinders in Equation (34) (Example 6): (a) plot and contour (n=5) and (b) plot and contour (n=7).

Example 7Consider a function originally studied by Lancaster and Salkauskas [10]

F(x, y)=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1 if (y−x)� 12

2(y−x) if 0�(y−x)� 12

12 [1+cos{4�[(x−1.5)2+(y−0.5)2]1/2}] if (x−1.5)2+(y−0.5)2� 1

16

0 otherwise

(35)

where 0�x�2 and 0�y�1. The function is piece-wise continuous having different types ofvariations in different regions of the domain. In this example the effect of the shape/interpolationfunctions obtained by using the Lagrange interpolation and the MLS interpolation on second-orderHDMR approximation is studied. Second-order HDMR approximation is studied by deployingseven (n=7), nine (n=9) and eleven (n=11) uniformly distributed sample points along the x- and



x

y

0.1

0.1

0.1

0.1

0.10.1

0.2

0.2

0.2

0.20.2

0.2

0.3

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.6

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9 0.9

1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

0

0.5

1

0

0.2

0.4

0.6

0.8

1

1.2

F (x

, y)

xy

(a) (b)

Figure 16. Function in Equation (35) (Example 7): (a) plot and (b) contour.

y-axes to form a regular grid in the ranges 0�x�2 and 0�y�1, and the reference point is taken asc=(0,0). Second-order HDMR approximation with n=7, 9 and 11 sample points involves 49, 81and 121 function evaluations, respectively. Figure 16 displays the plots and contours of the originalfunction.Figure 17(a)–(c) displays, respectively, the plots and contours of the approximate function by asecond-order HDMR with the MLS interpolation obtained using n=7, 9 and 11, respectively.Increase in the number of sample points resulted in reduction of the second-order HDMRapproximation error dramatically to 17.63, 14.43 and 5.42%, respectively, for n=7, 9 and 11. Theamount of error accumulated is due to the sudden peak and fall in the domain. With the increasein the number of sample points from n=7 to 11 second-order HDMR approximation is able tocapture the peak in the domain more accurately; however, the increase from n=7 to 11 resultedin oscillations in the flat regions.

Second-order HDMR approximation with n=7, 9 and 11 is repeated using the shape/interpolation functions obtained by the Lagrange interpolation. Figure 18(a)–(c) depicts the resultsobtained by second-order HDMR approximation, respectively, for the cases n=7, 9 and 11 usingthe Lagrange interpolation. Second-order HDMR approximation using the shape/interpolationfunction � j1 j2(xi1, xi2) obtained by the Lagrange interpolation resulted in the approximation errorof 55.09, 155.42, and 152.67%, respectively, for n=7, 9 and 11. The resulting approximationobtained using the Lagrange interpolation is far from the exact function and the approximationfails to portray the sudden peak and fall exhibited in the domain. Contrary to the expectation as thenumber of sample points is increased the approximation error is found to increase in second-orderHDMR approximation using the Lagrange interpolation. This demonstrates that the approximationusing the MLS interpolation scheme is superior to that of the Lagrange interpolation.

Example 8In this example a multi-dimensional surface described by a non-differential function of the form

F(x)=35−x21 −x22 −x3−x4−x5−x6− x7x8x9max(1, x10)

(36)



0 0.5 1 1.5 2

0

0.5

1

0

0.2

0.4

0.6

0.8

1

1.2

0

0

0

0

0

00

0

0

0

0

0

0

00

00

0

0

0

0

0

0

0

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.30.3

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F (x

, y)

F (x

, y)

F (x

, y)

x

y

0 0.5 1 1.5 2

0

0.5

1

0

0.2

0.4

0.6

0.8

1

1.2

-0.1

-0.1

-0.10

0

0

0

0

0

0

0

0

0

0

0

00

0

0

0

0 0

0

0

0

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Figure 17. First-order HDMR approximation of function in Equation (35) (Example 7): (a) plot andcontour (n=7); (b) plot and contour (n=9); and (c) plot and contour (n=11).



(c)

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F (x

, y)

F (x

, y)

F (x

, y)

Figure 18. Second-order HDMR approximation of function in Equation (35) (Example 7): (a) plot andcontour (n=7); (b) plot and contour (n=9); and (c) plot and contour (n=11).



Table I. Upper and lower bounds of variables x1−x10 in Example 8.

Variable

Bound x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

Upper −3.8 −3.8 1.3 −1.7 −1.7 −1.7 −2.0 0.18 0.918 −1.0Lower 3.4 3.4 3.7 6.7 6.7 6.7 4.0 2.28 1.050 5.0

is considered. The non-differentiability of the function arises due to x10 and the function ispiece-wise continuous at x10=1. The variables are assumed to be bounded as shown in Table I.First-order HDMR approximation is studied by deploying five (n=5) and seven (n=7) uniformlydistributed sample points along each of the variable axis in the range bounded by the upperand lower bounds of the variables x1−x10 given in Table I, and the reference point is taken asc=(0,0,0,0,0,0,0,0,0,0). First-order HDMR approximation with n=5 and 7 sample pointsinvolves 41 and 61 function evaluations, respectively. The computed approximation error usingEquation 28 for n=5 and 7, respectively, is 17.62 and 3.10%, which can be further reducedsignificantly by using second-order HDMR approximation. Second-order HDMR approximationwith n=5 and 7 produces the approximation error of 5.02 and 0.13%, respectively; however, thetotal number of function evaluations for second-order HDMR approximation with n=5 and 7 is,respectively, 761 and 1681, which is comparatively more than the number of function evaluationsfor first-order-approximation.

6. SUMMARY AND CONCLUSIONS

The potential of HDMR for tackling univariate and multivariate piece-wise continuous functionsis explored. An exact continuous function to replace a univariate or a multivariate piece-wisecontinuous function may not always be available in general problems. Rather than seeking an exactcontinuous function to replace a piece-wise continuous function, an equivalent continuous functioncan be found based on the HDMR concept. HDMR facilitates in an efficient manner the continuousfunction approximation accurately for univariate and multivariate piece-wise continuous functionsand to a reasonable accuracy even when the original function portrays the sudden peak and fallin the domain. First-order HDMR with more number of sample points or second-order HDMRapproximation results in dramatic reduction of the approximation error. Therefore, depending uponthe types of problem and its application, one needs to choose the number of sample points andorder of approximation in an optimum sense. It is also found that the approximation using theMLS interpolation scheme is superior to that of the Lagrange interpolation.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the financial support by the Board of Research in Nuclear Sciences,India (2004/36/39-BRNS/2332).

REFERENCES

1. Rabitz H, Alis OF, Shorter J, Shim K. Efficient input–output model representations. Computer PhysicsCommunications 1999; 117:11–20.



2. Rabitz H, Alis OF. General foundations of high dimensional model representations. Journal of MathematicalChemistry 1999; 25:197–233.

3. Alis OF, Rabitz H. Efficient implementation of high dimensional model representations. Journal of MathematicalChemistry 2001; 29(2):127–142.

4. Li G, Rosenthal C, Rabitz H. High dimensional model representations. Journal of Physical Chemistry A 2001;105:7765–7777.

5. Li G, Wang SW, Rabitz H. High dimensional model representations generated from low dimensional datasamples—I. mp—Cut—HDMR. Journal of Mathematical Chemistry 2001; 30:1–30.

6. Wang SW, Levy II H, Li G, Rabitz H. Fully equivalent operational models for atmospheric chemical kineticswithin global chemistry-transport models. Journal of Geophysical Research 1999; 104(D23):30417–30426.

7. Li G, Wang SW, Rabitz H, Wang S, Jaffe P. Global uncertainty assessments by high dimensional modelrepresentations (HDMR). Chemical Engineering Science 2002; 57(21):4445–4460.

8. Sobol IM. Theorems and examples on high dimensional model representations. Reliability Engineering andSystem Safety 2003; 79:187–193.

9. Balakrishnan S, Roy A, Ierapetritou MG, Flach GP, Georgopoulos PG. A comparative assessment of efficientuncertainty analysis techniques for environmental fate and transport models: application to the FACT model.Journal of Hydrology 2005; 307(1–4):204–218.

10. Lancaster P, Salkauskas K. Curve and Surface Fitting: an Introduction. Academic Press: London, 1986.11. Singh IV. A numerical solution of composite heat transfer problems using meshless method. International Journal

of Heat and Mass Transfer 2004; 47:2123–2138.12. Rao BN, Rahman S. An efficient meshless method for fracture analysis of cracks. Computational Mechanics

2000; 26:398–408.


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High dimensional model representation for piece-wise continuous function approximation