a new sequential optimal sampling method for radial basis functions

Applied Mathematics and Computation 218 (2012) 9635–9646

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

A new sequential optimal sampling method for radial basis functions

Xin Wei, Yi-Zhong Wu ⇑, Li-Ping ChenNational CAD Supported Software Engineering Centre of Huazhong, University of Science and Technology, Wuhan, Hubei, PR China

a r t i c l e i n f o a b s t r a c t

Keywords:MetamodelSequential samplingRadial basis functionsModel approximation

0096-3003/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.amc.2012.02.067

⇑ Corresponding author.E-mail address: [email protected] (Y.-Z.

The sampling procedure is crucial for accuracy of model approximation for a given meta-model. In this paper, we propose a new sequential sampling method for radial basis func-tions. At each sampling step, we provided a criterion to determine the optimal samplingpoint, which maximizes the value of the product of curvature and square of minimum dis-tance to other design sites. Experiments proved that using the same sampling size thehighest accuracy and efficiency can be obtained to approximate an objective functionunder this criterion.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

In the past 2 decades, the use of metamodel, also called response surface methods (RSM), surrogates, have attracted inten-sive attention. A metamodel is an approximation to system response constructed from its value at a limited number of se-lected input values, the design of experiments (DoE). It is found to be a valuable tool to support a wide scope of activities inmodern engineering design, especially design optimization. As one of the most effective metamodel, radial basis functions(RBF) interpolation have been gained popularity for model approximation because of their simplicity and very accurate re-sults for interpolation problems. RBF is becoming a viable choice for finding the global optimum of computationally expen-sive functions.

The sampling method is one of the most important factors of affecting accuracy for a given metamodel [1]. Classic sam-pling methods were developed from DoE. These methods focused on planning experiment and tend to spread the samplepoints around boundaries of the design space for eliminating random error. The classical experimental designs include fac-torial or fractional factorial design [2], central composite design (CCD) [2,3], Box–Behnken [2]. However, Sacks et al. [4] sta-ted that classic designs can be inefficient or even inappropriate for deterministic optimal problems. Jin et al. [5] confirmedthat experiments designs for deterministic computer analyses should be spaced filling. Space-filling designs include grids,Latin Hypercube designs (LHD), orthogonal arrays [2].

Because of the difficulty of forecasting the number of sampling points, sequential sampling has recently has gained pop-ularity for its advantages of flexibility and adaptability of over other methods. Jin et al. [6] stated that sequential samplingallows engineers to control the sampling process and it is generally more efficient than one-stage sampling. Lin [7] proposeda sequential exploratory experiment design (SEED) method for Kriging models. Sasena et al. [8] used the Bayesian predictingmethod to adaptively identify sample points. However, all of mentioned sampling methods are common one-stage methodsfor most of metamodel or specific methods only for some metamodel except RBF. So far there is no efficient sequential sam-pling method for RBF.

In this paper, we propose a new sequential sampling method which named ‘‘maximum curvature and minimum pointdistance based sequential sampling method (MCMPD)’’. In order to utilize the geometrical feature of metamodel, maximum

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http://dx.doi.org/10.1016/j.amc.2012.02.067

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9636 X. Wei et al. / Applied Mathematics and Computation 218 (2012) 9635–9646

curvature of the response surface and minimum distance among the sampling sites, as a general sampling criterion, are ap-plied in sequential sampling procedure. For the simplicity of RBF model, we can easily evaluate curvature on design opti-mum. A new model approximation algorithm integrated sequential optimal sampling is presented. To illustrate theaccuracy and efficiency of the proposed algorithm, the measure performance and several test problems will be provided.

2. Sequential optimal sampling method for RBF

2.1. Radial basis functions

The RBF metamodel was originally developed by Hardy [9] in 1971 to fit irregular topographic contours of geographicaldata. Dyn et al. [10] made radial basis functions more practical by enabling them to smooth data as well as interpolate it.Duchon [11] added a polynomial to the definition of RBF for improving the performance of thin-plate spline basis function.Wu [12] provided a compactly support radial basis functions which produce series of positive definite radial function.

Given N sampling data S : [xi,yi] (i = 1,2, . . . ,N), where xi is p-dimensional vector (i.e. sampling site) and yi is its correspond-ing real response value, the standard RBF model of expression (1) has the general form of

y ¼ f /ðxÞ ¼PNi¼1

ki/ðkx� xikÞ ¼ U � k; ð1Þ

where U = [/1,/2, . . . ,/N] (/j = /kx � xjk), k = [k1,k2, . . . ,kN]T, x is a vector of design variables, xi is a vector values of designvariables at the ith sampling point, kx � xik is the Euclidean norm, / is a basis function, and ki is the coefficient for the ithbasis function. The approximation function y is actually a linear combination of some RBFs with weight coefficients. Themost commonly-used basis functions are listed as following:

Linear
/(r) = r Cubic /(r) = (r + c)3
Thin-plate spline
/(r) = r2ln (cr) Gaussian /ðrÞ ¼ e�cr2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip Multiquadric /ðrÞ ¼ r2 þ c2
Inverse-multiquadric
/ðrÞ ¼ 1ffiffiffiffiffiffiffiffiffir2þc2p
where c is a constant, and 0 < c 6 1.Now with the given N sample data, we can determine the N coefficients ki through solving the following linear equations:

Ak ¼ y) k ¼ A�1y; ð2Þ

where A is the design matrix, Aij = /(kxi � xjk), y is the vector: y = [y1,y2, . . . ,yN]T.However, the radial function is only conditionally positive definite in some case such as the thin-plate spline radial basis

function. To deal with this situation, an RBF model can be augmented with a linear polynomial given as

y ¼ f /ðxÞ ¼PNi¼1

ki/ðkx� xikÞ þPpj¼1

cjgjðxÞ; ð3Þ

where g(x) is a linear polynomial function, p is the total number of terms in the polynomial, and cj (j = 1,2, . . . ,p) are the un-known coefficients. Eq. (3) is underdetermined, because the number of parameters to be solved is more than that of theequations created with available data points. Therefore, the orthogonality condition is further imposed on coefficients kso that

Pni¼1

kigiðxiÞ ¼ 0:

The accuracy of the RBF model depends on two factors, the sampling strategies and the choice of basis functions. Simpsonet al. [1] found that the type of DoE and samples size will affect the accuracy of the RBF model. Buhmann [13] state that thin-plate spline turned out to be excellent for mapping of images; multiquadric approximations are performing well for spher-ical surface problems. Fang [14] stated that the augmented RBF models created by Wu’s compactly supported functions arethe most accurate for the various test functions. This paper will not focus on the different form of RBF model but the strat-egies of sampling method, which means that the proposed method can be used for all forms of RBF models.

2.2. The new sequential sampling method

As it is known, the optimal sampling site is the position where the error between the real value and the predict valuereaches highest. General knowledge tells us that sampling at the site with the maximum curvature of the response surface

X. Wei et al. / Applied Mathematics and Computation 218 (2012) 9635–9646 9637

would improve the accuracy at the greatest extent. The curvature is a measure of how ‘curved’ a curve is. The geometricdescription of curvature as shown in Fig. 1.

In mathematics, the curvature K of a is the rate of change of direction at that point of the tangent line with respect to arclength. Let the curve C is smooth, the selected point M0 on the curve C as a basis points of the arc s. Set point M on the curve Ccorresponding to the arc s, the tangent of angle of inclination is a, another point M0 on the curve C corresponding to the arcs + Ds, the angle of inclination of tangent is a + Da. So the arc length of is jDsj, and when the fixed point moves from M to M0,the tangent of the angle turned to jDaj. The curvature of point M on the curve C can be defined as

K ¼ limDs!0

DaDs

��:

However, only under this criterion the sampling sites must gather and converge into some identical and reduplicate sam-pling points which will make RBF model singular. So we need consider the impact of distance among the sampling sites. Infact, we can adopt the criterion:

find : x;

max : KðxÞ � dDminðxÞ;

ð4Þ

where dmin(x) is minimum distance from x to other existing sampling sites, K(x) is the curvature of the site x on the responsesurface. The power D can be set to 1–2 according to test experience in Sections 3.2 and 3.3. In general, we can set D = 2 for 1or 2 variables problem; if the number of variables greater than 2, D = 1. Because the criterion utilizes the combination ofmaximum curvature and minimum point distance, so we named the new method as ‘‘maximum curvature and minimumpoint distance based sequential sampling method’’.

Considering the RBF model with p dimensions in Eq. (1), the grade of the equation is

@ f@x¼PNi¼1

ki/0ðriÞ

riðxÞðx� xiÞT ; ð5Þ

where ri(x) = kx � xik, /0(ri) is the partial derivative of /(ri) respect to ri. For linear RBF model, /(ri) = ri, then /0(ri) = 1.Then we can get the Hessian matrix as

HðxÞ ¼ @2 f@x2 ¼

PNi¼1

ki

riðxÞ/0ðriÞ � I þ /00ðriÞ �

/0ðriÞriðxÞ

� �� ðx� xiÞT : ð6Þ

Similarly, /00(ri) is the partial derivative of /0(ri) respect to ri. For linear RBF model, /(ri) = ri, then /00(ri) = 0. I is unit vector inEq. (6).

For a multiple dimensional problem, K(x) can be measured by the Eigen values of Hessian matrix of the response surfaceaccording to the principal curvature in differential geometry. Generally, for the Hessian matrix of (5) we can get its p Eigenvalues k1,k2, . . . ,kp. To this end, the curvature of x at the response surface can be calculated as

KðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPpi¼1

kiðxÞ2s

: ð7Þ

As we know, if the local area near x is a super plane, then K(x) = 0. Now we add a constraint to K(x),

KðxÞP e ð8Þ

which means that if K(x) 6e, set K(x) = e, e is a little positive real number like 10�8.

Theorem 1. For an arbitrary black-box original function, if we sample under the criterion (4) with the constraint (8), then the localarea near any given site P0 can be sampled as the size of sampling size reaches to infinite.

α α α+ Δ

C

αΔ

0M ssΔ

M

'M

xo

y

Fig. 1. Geometric description of curvature.


Proof. Suppose the minimum distance to other sites and the curvature near site P0 are d0 and K0 (Pe) respectively, thesequence of the sampling sites are P1,P2, . . . ,PN. With increasing of the sampling size N; KNdD

N becomes smaller and smaller,tending to zero. But the criterion at P0 is a constant K0dD

0 (> 0), for sure there exists a sampling size N which satisfiesK0dD

0 P KNdDN .

Then the local area near P0 will be sampled at the next step. Proof is completed. h

Before sequential sampling, the MCMPDS method will generate a lot of initial sampling points uniformly distributed inthe design space. Theorem 1 guarantees that local area near arbitrary point can be sampled, when initial sampling pointsfill the whole design space, then the area near these points will be sampled and the entire domain of black-box function willbe covered. Hence, the response surface will be accurate adequately when the sampling points become more and more. Espe-cially, it guarantees that all peaks or valleys of the original function can be sampled when the sampling size is large enough.

For example in Fig. 2, at the beginning, the sampling sites may be centered on B (as Fig. 2(a) shows). With the samplingcontinuing, the sites near B becomes denser and denser (as Fig. 2(b) shows). Ultimately the sampling sites will be centered onA (as Fig. 2(c) shows).

Now we can predict that without constraint (8) the valley A in Fig. 2 could be lost even if the sampling size is infinitelylarge.

2.3. Multi-dimensional global approximation algorithm

The flowchart of the proposed multi-dimensional global approximation (MGA) algorithm integrated MCMPDS criterion isshown in Fig. 3.

Step 1: Initial sampling. Unlike the global optimization algorithm using LHD usually, here we adopt grid sampling withidentical levels q. The range of q is q = 2 � 6 according to p and the predicted fairness of the original function. Ingeneral, we set q = 3. If p P 3, set q = 2. If q is an even number, we append the center as well.

Step 2: Construct initial RBF model. Obtain the responses of the initial sampling sites through evaluating the originalfunction (usually executing a computation-intensive analysis or a black-box simulation), then construct the ini-tial RBF model with these sampling data.

Step 3: Check stopping criteria. The off-design test like RSME analysis needs too many function evaluations, so it is notsuitable for stopping criteria. In this paper, two criteria are adopted at this step: one is the maximum number ofiterations, M1; the other is the maximum number of continuous invalid sampling points, M2. The former limitsthe number of function evaluations while the latter means that the model is accurate enough. The continuousinvalid sampling points is the successive sampling points [xi,yi] (i = S + 1,S + 2, . . . ,S + M2, S is an integer) whichmeet

jyi � yi�1j 6 da orjyi � yi�1j

�yi6 dr

where yi is the real response value of a sampling site xi; yi�1 is the approximate value of the RBF model con-structed without the point [xi,yi]; �yi is the mean value of all the existing responses values; da and dr are the abso-lute tolerance and relative tolerance respectively, a positive number set by user. If one of the criteria is satisfied,go to Step 6.

Step 4: Sequential optimal sampling. The optimal sampling site is searched on the current response surface which meetsthe criterion (3) with constraint (7). This is a typical global optimization problem which can be solved by theDIRECT [15] algorithm.

A B

A

A

B

B

(a)

(b)

(c)

Fig. 2. Example for sequential optimal sampling.

Fig. 3. The flowchart of MGA algorithm.


Step 5: Update the RBF model. Execute evaluating at the optimal site, then update the RBF model and refresh the infor-mation of the RBF model structure. Go to Step 3.

Step 6: Stop.

3. Tests of the MGA algorithm

3.1. Measure performance

� RMSEGenerally speaking, a RBF response surface passes through all the sampling points exactly. So it is impossible to estimatethe accuracy of a RBF model with ANOVA. We can use additional testing points to evaluate the accuracy of the model. TheRMSE (root of mean square errors) value for the testing points is calculated by the following equation:

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1ðfi � f iÞ2

n

s;

where n is the number of testing points, fi and f are the true function value and predicted value calculated from the RBFmodel at the ith testing point respectively. RMSE is used to gauge the overall accuracy of the model. The smaller the valueof RMSE, the better accurate the RBF response surface will be.� Invalid sampling points (ISP)

If a sample point can be removed without affecting the accuracy of RBF model, we can say that the point is invalid sam-pling point. When the invalid sampling points are removed, the remainder sites are valid points. The more invalid points,the more invalid function evaluations occur, which means a waste of valuable computing resources. Hence, for the samesize of sampling sites, the more the sampling data has invalid sampling points, the worse the efficacy of sample method is.We also can use the number of the invalid sampling points to evaluate its efficacy roughly and quickly without additionaltesting points.Considering the RBF model using N sampling points as the equation (1) shows, now we add a new distinct pointPN+1([xN+1,yN+1]) to update the RBF model. If the PN+1 is on the old RBF response surface, then the Eq. (1) becomes


/ /0Nþ1

/Nþ1 0

!k

kNþ1

� �¼

y

yNþ1

� �;

which is equivalent to

/kþ /0Nþ1kNþ1

/Nþ1k

!¼

y

yNþ1

� �:

Because /k = y and /0Nþ1 – 0, that mean the unknown parameters kN+1 is equal to zero. In practice, the size of coefficientreflects the importance of corresponding point. The smaller the coefficient, the less the point contributes to accuracy ofRBF model. The invalid point results in the corresponding coefficient equals or close to 0. Contrarily, we can get the invalidsampling points from an existing RBF model according the coefficients of the model.

For the RBF model in (3), suppose

�k ¼ 1NPNi¼1jkij;

for an arbitrary sampling point Pj(xj,yj), if

jkjj 6 �k � 5%;

we can say that Pj is an invalid sampling point, the remainder points are valid points.To prove our hypothesis, an example of invalid sampling points is given as Fig. 4. Fig. 4(a) is the figure of Peaks function

approximated by a RBF model with grid sampling size 15 � 15; Fig. 4(b) is the RBF figure after removing 47 invalid samplingpoints. As the figure shows, the accuracy remains almost the same before and after removing invalid sampling points. TheRSME test also supports the judgment:

RSMEðaÞ ¼ 0:1041; RSMEðbÞ ¼ 0:1055:

RSME(a) and RSME(b) are RSME of model in Fig. 4(a) and (b) respectively. Usually, the off-design tests are executed throughLHD sampling. With respect to the randomness, the RSME is the mean value of multiple tests with the sampling size is verylarge at each test.

3.2. Numerical test

Now we will test the MGA algorithm using some well-known benchmark functions. All these functions are approximatedwith RBF model, but the response surfaces are constructed through three different sampling strategies (i.e. LHD, grid andMCMPDS) with same sampling size. The ISP (invalid sampling points) and RSME (through additional off-design testing) ofthe three strategies will be compared.

3.2.1. Peaks function

y ¼ 3ð1� x1Þ2 � e�x21�ðx2þ1Þ2 � 10

x1

5� x3

1 � x52

� �� e�x2

1�x22 � 1

3e�ðx1þ1Þ2�x2

2 ; ½x1; x2� 2 ½�3;�4; 3;4�:

Fig. 4. An example of invalid sampling points.


Setting D = 2, q = 3, M1 = 91(N = 100), M2 = 5, dr = 1%, the response surfaces of the RBF models via different samplings areshown in Fig. 5, and Table 1 compares the approximate results at two aspects: number of invalid sampling points (#ISP)and RSME.

3.2.2. Easom function

2 2
y ¼ � cos x1 � cos x2 � e�ðx1�pÞ �ðx2�pÞ ; ½x1; x2� 2 ½�10;�10; 10;10�
Setting D = 2, q = 3, M1 = 55 (N = 64) and M2 = 5, d = 1%, the response surfaces of the RBF models via different samplings areshown in Fig. 6, and Table 2 compares the approximate results at two aspects: number of invalid sampling points (#ISP) andRSME.

3.2.3. Golden-price function

2
y ¼ ð1þ ðx1 þ x2 þ 1Þ ð19� 14x1 þ 3x21 � 14x2 þ 6x1x2 þ 3x2
2ÞÞ

� ð30þ ð2x1 � 3x2Þ2 18� 32x1 þ 12x21 þ 48x2 � 36x1x2 þ 27x2

2

� � �; ½x1; x2� 2 ½�2� 2; 22�

Setting D = 2, q = 3, M1 = 91 (N = 100) and M2 = 5, d = 1%, the response surfaces of the RBF models via different samplings areshown in Fig. 7, and Table 3 compares the approximate results at two aspects: number of invalid sampling points (#ISP) andRSME.

3.2.4. Schaffer’s function ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq� �
y ¼ 0:5þ
sin2 x21 þ x2

2 � 0:5

1þ 0:001 x21 þ x2

2

� � ; ½x1; x2� 2 ½�2� 2; 22�:

Fig. 5. RBF models of Peaks function using different sampling strategies.

Table 1Comparison of the RBF models for peaks function.

LHD Grid MCMPD

#ISP 19 16 5RSME 0.5369 0.3200 0.1643

Fig. 6. RBF models of Easom function using different sampling strategies.

Table 2Comparison of the RBF models for Easom function.

LHD Grid MCMPD

#ISP 40 29 14RSME 0.0397 0.0378 0.0064


Setting D = 2, q = 3, M1 = 55 (N = 64) and M2 = 5,d = 1%, the response surfaces of the RBF models via different samplings areshown in Fig. 8, and Table 4 compares the approximate results at two aspects: number of invalid sampling points (#ISP) andRSME.

3.2.5. Hartmann function

" #
y ¼
P4i¼1

ai exp �P3j¼1

Aijðxj � PijÞ2 ; a ¼ ½1;1:2;3;3:2�T ;

A ¼

3:0 10 300:1 10 353:0 10 300:1 10 35

0BBB@

1CCCA; P ¼ 10�4

3689 1170 26734699 4387 7471091 8732 5547381 5743 8828

0BBB@

1CCCA:

Setting D = 1, q = 2, M1 = 115 (N = 125) and M2 = 5, d = 1%, the Table 5 compares the approximate results at two aspects: num-ber of invalid sampling points (#ISP) and RSME.

3.3. Design problems

3.3.1. Tension–compression string problemThe tension–compression string design problem is firstly proposed by Arora [16] and the aim is to minimize the weight of

a tension–compression spring (Fig. 9) subject to constraints on minimum deflection, shear stress, surge frequency, limits on

Fig. 7. RBF models of Golden-price function using different sampling strategies.

Table 3Comparison of the RBF models for Golden-price function.

LHD Grid MCMPD

#ISP 9 9 8RSME 2.0746e + 4 6.3349e + 03 5.6368e + 3


outside diameter and on design variables. The design variables are the mean wire diameter d, the coil diameter D, and thenumber of active coils N.

The problem can be stated as

min f ðxÞ ¼ ðx3 þ 2Þx2x21;

Subjected to

g1ðxÞ ¼ 1� x32x3

71785x41

6 0;

g2ðxÞ ¼4x2

2 � x1x2

12566ðx2x31 � x4

1Þþ 1

5108x21

� 1 6 0;

g3ðxÞ ¼ 1� 140:45x1

x22x3

6 0;

g4ðxÞ ¼x2 þ x1

1:5� 1 6 0;

where

x ¼ ðx1; x2; x3ÞT ¼ ðd;D;NÞT :

Setting D = 1, q = 2, M1 = 17 (N = 27) and M2 = 5, d = 1%, the Table 6 compares the approximate results at two aspects: numberof invalid sampling points (#ISP) and RSME.

Fig. 8. RBF models of Schaffer’s function using different sampling strategies.

Table 4Comparison of the RBF models for Schaffer’s function.

LHD Grid MCMPD

#ISP 8 2 0RSME 0.0749 0.0281 0.0228

Table 5Comparison of the RBF models for hartmann function.

LHD Grid MCMPD

#ISP 4 6 3RSME 0.9062 0.7301 0.6623

Fig. 9. Tension compression string figure.


3.3.2. Welded beam problemThe welded beam problem design problem is firstly proposed by Ragsdell and Phillips [17]. The welded beam assembly is

shown in Fig. 10. The system consists of the beam A and the weld required securing the beam to the member B. This design

Table 6Comparison of the RBF models for tension–compression string.

LHD Grid MCMPD

#ISP 4 0 3RSME 1.2070 14.3713 0.6982

Fig. 10. Welded beam design figure.

Table 7Comparison of the RBF models for welded beam.

LHD Grid MCMPD

#ISP 3 16 6RSME 179.475 287.8146 51.1709


problem deals with the minimization of cost and the end deflection of a beam welded at one end and carrying a loadF = 6000 lb at the other. The design variables are the thickness of the beam b, width of the beam t, length of the weld land weld thickness h. The overhang portion of the beam has a length of 14 in. The constraints are the maximum allowablestress in the weld, the maximum bending stress in the bar, the buckling load being less than the critical value and the deflec-tion of the end of the beam being less than a designated limit.

The problem can be stated as

min f ðxÞ ¼ 1:10471h2lþ 0:04811tbð14:0þ lÞ;

subjected to

g1ðxÞ ¼ 13600� sðxÞP 0;g2ðxÞ ¼ 30000� rðxÞP 0;g3ðxÞ ¼ b� h P 0;g4ðxÞ ¼ PcðxÞ � 6000 P 0;

where

x ¼ ðx1; x2; x3; x4ÞT ¼ ðh; l; t; bÞT ;

sðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs0Þ2 þ ðs00Þ2 þ ðls0s00Þ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:25ðl2 þ ðhþ tÞ2Þ

qr;

s0 ¼ 6000ffiffiffi2p

hl;

s00 ¼6000ð14þ 0:5lÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:25ðl2 þ ðhþ tÞ2

q2½0:707hlðl2=12þ 0:25ðhþ tÞ2Þ�

;

rðxÞ ¼ 504000t2b

;

PcðxÞ ¼ 64746:022ð1� 0:0282346tÞtb3:

Setting D = 1, q = 2, M1 = 64 (N = 81) and M2 = 5, d = 1%, the Table 7 compares the approximate results at two aspects: numberof invalid sampling points (#ISP) and RSME.


4. Discussions and conclusions

The results of the examples in Section 3 show that, the accuracy of RBF model using MCMPDS method is higher than othermethods. It can be observed intuitionally that the curved surface of the RBF approximation adopting MCMPDS method issmoother than LHD and Grid methods. From the Tables above, the RSME of RBF models using MCMPD methods is smallerthan other methods. The examples above also indicate that the proposed approach has been higher efficiency for mostRBF model, especially the two-dimensional RBF model. The MCMPD method can reduce more invalid point compared withthe LHD and Grid methods, which means need much less number of function evaluations in the model approximation.

The new sequential optimal sampling method which sample sites step by step make the user can easily control the pro-cess of model approximation on demand, and the procedure of sampling points can be stopped at any time, which couldreduce expensive computations as far as possible. Therefore, a lower computational cost would be expected when the pro-posed method is employed.

A maximum curvature and minimum point distance based sequential sampling method has been proposed, which pro-vides another effective way for constructing model approximation. It is superior in sampling efficiency compared with thosemethods involving LHD and Grid, and meanwhile, could improve the accuracy of the RBF model. Therefore, the sequentialoptimal sampling method significantly outperforms the conventional sampling technique.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 51175198).

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a new sequential optimal sampling method for radial basis functions

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