a new multiquadric quasi-interpolation operator with interpolation property

Research Article

Received 28 August 2012 Published online 19 July 2013 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2915MOS subject classification: 65D05; 65D10; 65D15

A new multiquadric quasi-interpolationoperator with interpolation property

Jinming Wu*†

Communicated by T. Qian

In this article, we discuss a class of multiquadric quasi-interpolation operator that is primarily on the basis ofWu–Schaback’s quasi-interpolation operator LD and radial basis function interpolation. The proposed operator possessesthe advantages of linear polynomial reproducing property, interpolation property, and high accuracy. It can be appliedto construct flexible function approximation and scattered data fitting from numerical experiments. Copyright © 2013John Wiley & Sons, Ltd.

Keywords: radial basis function; multiquadric function; quasi-interpolation; interpolation property

1. Introduction

A radial basis function (RBF) is a relatively simple multivariate function generated by a univariate function. RBF method provides excel-lent interpolants for high dimensional scattered data. Due to its simple form and good approximation behavior, the RBF methodhas become an effective tool in a number of fields such as multivariate function approximation, neural networks, and the numericalsolutions of differential equations during the last two decades [1–6].

The process of RBF interpolation is as follows: For a given region � � Rd and a set of distinct points X D fxigNiD1 � � and the

corresponding function values ff .xi/gNiD1, we can construct an interpolant to a given function f of the form

PX f .x/DNX

jD1

cj�.kx � xjk2/C

MXkD1

dkpk.x/, (1)

where k � k2 denotes the Euclidean norm, � : RC ! R is a given RBF, and p1.x/, � � � , pM.x/ form a basis for …dm.x/ that denotes the

linear space of polynomials of total degree at most m in d variables.The coefficients cj and dk can be determined by solving the linear system

NXjD1

cj�.kxi � xjk2/C

MXkD1

dkpk.xi/D f .xi/, iD 1, � � � , N,

NXjD1

cjpl.xj/D 0, 1� l �M.

(2)

It is obvious that the addition of polynomials of total degree at most m guarantees polynomial precision, that is, if the data comefrom a polynomial of total degree at most m, they are fitted by that polynomial. From the papers [4, 5], we know that this interpola-tion problem (2) is well-posed if �.r/ is a conditionally positive definite RBF of order m. If m D 0, �.r/ is said to be strictly positivedefinite RBF.

In the summarized paper [6], Franke pointed out that multiquadric (MQ) interpolation was best among 29 scattered data interpo-lation methods in terms of timing, storage, accuracy, visual pleasantness of surface, and ease of implementation. Although the MQinterpolation (with appended constant) is always solvable, the resulting matrix from using MQ quickly becomes ill-conditioned as the

Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China*Correspondence to: Jinming Wu, Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China.†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014, 37 1593–1601

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number of points increases. Thus, the research focus on the MQ quasi-interpolation. MQ quasi-interpolation is constructed directly fromlinear combination of MQ basis and the function values at the sites. In the early 1992, Beatson and Powell [7] proposed three univariateMQ quasi-interpolation operators, namely, LA, LB , and LC . Because LC requires the derivative values at the endpoints, it is not con-venient for practical use. Later, Wu and Schaback [8] proposed the univariate MQ quasi-interpolation LD without using the derivativevalues at the endpoints and proved it can have an O.h2j log hj/ error only if at least the shape parameter c D O.h/. In 2004, Ling [9]proposed a multilevel quasi-interpolation operator LR and proved that it converges with a rate of O.h2.5j log hj/ as cDO.h/. In 2009,Chen et al. [10] developed a MQ quasi-interpolation f�.x/, which has the properties of linear reproducing and preserving monotonicity.Feng and Li [11] constructed a shape-preserving quasi-interpolation operator by shifts of cubic MQ functions and proved it can pro-duce an error of O.h2/ as cDO.h/. In 2010, Wang et al. [12] proposed a kind of improved univariate MQ quasi-interpolation operatorsLH2m�1 by using Hermite interpolating polynomials, and the convergence rate depended heavily on the shape parameter c. Jiang etal. [13] proposed two new multilevel univariate MQ quasi-interpolation operators LW and LW2 with higher approximation order. Veryrecently, Gao and Wu [14] studied the quasi-interpolation for the linear functional data rather than the discrete function values.

Meanwhile, MQ quasi-interpolation has been used in a wide range of fields successfully. For example, in 2007, Wang and Wu [15]applied operator LD to tackle approximate implicitization of parametric curves. In 2011, Wu [16] presented an new approach to con-struct the so-called shape preserving interpolation curves based on MQ quasi-interpolation LD . Hon and Wu [17], Wu [16], Chen andWu [18, 19], Jiang and Wang [20], and other researches provided some successful examples using MQ quasi-interpolation operators tosolve different types of partial differential equations.

In this paper, we mainly construct a new MQ quasi-interpolation operator based on Wu–Schaback’s quasi-interpolation operator LDand RBF interpolation. The rest of this paper is organized as follows. In Section 2, the operator LD and its basic properties are reviewed.In Section 3, we present a new quasi-interpolation operator LG . Its main properties and accuracy are given in Section 4. Meanwhile, weextend the proposed operator to higher dimension in Section 5. Finally, we give several practical applications in Section 6.

2. Multiquadric quasi-interpolation operator LD

The univariate MQ quasi-interpolation of a function f : Œa, b�!R at the scattered points

aD x0 < x1 < � � �< xn D b,

has the form

Lf .x/ :DnX

iD0

f .xi/ i.x/,

where each i.x/ is the linear combination of the MQ basis introduced by Hardy [21]

�i.x/Dq.x � xi/2C c2

and c is a shape parameter. In [8], Wu–Schaback’s MQ quasi-interpolation operator LDf .x/ is defined as follows

LDf .x/DnX

iD0

f .xi/ i.x/, x 2 Œa, b�, (3)

where

0.x/D1

2C�1.x/� .x � x0/

2.x1 � x0/,

1.x/D�2.x/� �1.x/

2.x2 � x1/��1.x/� .x � x0/

2.x1 � x0/,

i.x/D�iC1.x/� �i.x/

2.xiC1 � xi/��i.x/� �i�1.x/

2.xi � xi�1/, iD 2, � � � , n� 2,

n�1.x/D.xn � x/� �n�1.x/

2.xn � xn�1/��n�1.x/� �n�2.x/

2.xn�1 � xn�2/,

n.x/D1

2C�n�1.x/� .xn � x/

2.xn � xn�1/.

LDf .x/ has the following properties and error estimate (see [8] for details).

Theorem 2.1If the data ffig

niD0 stem from a convex (concave, linear) function, then the quasi-interpolation operator LDf is convex (concave, linear)

function.

Theorem 2.2MQ quasi-interpolation operator LDf is monotonicity preserving.

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Theorem 2.3For f 2 C2Œa, b�, the quasi-interpolant LDf satisfies an error estimate of type

kf �LDfk1 ��

C1h2C C2chC C3c2 log h�kf 00k1

with suitable positive constants C1, C2, and C3, independent of h and c. Here,

hD max1�i�n

jxi � xi�1j.

3. A new multiquadric quasi-interpolation operator LG

It is noted that we may obtain high approximation order by using RBF interpolation, but we have to solve an unstable linear systemof equations when the number of interpolation centers increases. The MQ quasi-interpolation operator can avoid this problem. So weconstruct a new MQ quasi-interpolation operator denoted by LG f .x/ that is based on RBF interpolation at a given set a relatively small

number of interpolated data and quasi-interpolation operator LDf .x/ at another set of relatively large number of approximated data.The most important thing is that LG f .x/ does not require any derivative values of f .x/ at scattered data points compared with theexisting operators in [11–13].

For a given set of scattered data X :D fxigniD0, xi 2 Œa, b�, we obtain the MQ quasi-interpolation operator LDf .x/ that is defined by

(3) on the data .xi , f .xi//niD0, that is,

LDf .x/DnX

iD0

f .xi/ i.x/.

Then, an error function is defined in the form as follows

E.x/D f .x/�LDf .x/. (4)

Meanwhile, another set of points X D˚

xki

�NiD0 � X is introduced and the corresponding function values

˚f .xki /

�NiD0 are known,

where n is far less than n and 0� k0 < k1 < � � �< kN � n. The filled distance is defined as

hX ,� D supx2�

minxki2Xjx � xki j. (5)

Without loss of generality, we can obtain a RBF interpolation function with compactly supported radial basis function (CSRBF) �.r/[22] on the set X and denoted by

PX E.x/DNX

iD0

ci�˛�jx � xki j

�, �˛

�jx � xki j

�D �

�jx � xki j

˛

�(6)

satisfying

PX E�

xkj

�D E

�xkj

�, jD 0, 1, � � � , N, (7)

where ˛ is the support size of �.r/.It is known that the coefficients ci , iD 0, 1, � � � , N can be determined by the linear system

AC D E,

where,

AD��˛

�jxki � xkj j

�, C D .c0, c1, � � � , cN/

T ,

E D�E�

xk0

�, E�

xk1

�, � � � , E

�xkN

�T.

Therefore, our MQ quasi-interpolation operator is given by

LG f .x/D LDf .x/CPX E.x/. (8)


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4. Its main properties and accuracy

In this section, we give its properties and accuracy of quasi-interpolation operator LG f .

Theorem 4.1MQ quasi-interpolation operator LG f satisfies linear polynomial reproducing property.

ProofIf f .x/D axC b, then LDf .x/D f .x/. So, E.x/D 0 and PX E.x/D 0. We obtain LG f .x/D LDf .x/CPX E.x/D f .x/. �

Theorem 4.2MQ quasi-interpolation operator LG f satisfies interpolation property on the set X .

ProofObviously,

LG f�

xki

�D LDf

�xki

�CPX E

�xki

�D LDf

�xki

�C f

�xki

��LDf

�xki

�D f

�xki

�, iD 0, 1, � � � , N.

�

Theorem 4.2 tells us that the quasi-interpolation operator LG f can interpolate arbitrary set of scattered data as we require to inter-polate previously. That is to say, X may not be a subset of X and can be completely different from X . This property is very useful in thefields of curve/surface design and modeling. Let us explain this case explicitly. For example, let X D fx0, x1, � � � , xNg be differ from X .Firstly, we compute

E.xi/D yi �LDf .xi/, iD 0, 1, � � � , N,

where, yi is the corresponding function value at the site xi and it therefore should be given or known in advance.Secondly, we construct the interpolation function

PX E.x/DNX

iD0

ci�˛.jx � xij/

and it satisfies the following linear system

PX E.xj/D E.xj/, jD 0, 1, � � � , N.

Thus, we can obtain the interpolation function PX E.x/ and its MQ the quasi-interpolation operator LG f .x/D LDf .x/CPX E.x/.However, our quasi-interpolation scheme LG f .x/ can approximate the set of scattered data X well and have the interpolation prop-

erty on the set X , so it is reasonable and convenient that X is chosen to be a subset of the set X . At least, X is required to be closer tothe set X .

According to the error estimate of RBF interpolation by using Wendland’s CSRBF, we give the accuracy of the quasi-interpolation LG f .The error estimate of the RBF interpolation is considered to be taken in the native space N� determined by RBF �, where N� is definedto be the completion of preHilbert space

H� D spanf�.., y/ : y 2�g.

Theorem 4.3 ([23])Let�k denote the CSRBF with smoothness 2k of minimal degree. The native spaceN� coincides with Sobolev space Hs.R/with sD kC1and the native space norm is equivalent to the Sobolev norm.

Theorem 4.4 ([23])For every u.x/ 2 Hs.R/, then the interpolant Pf ,X satisfies the estimate error

ku.x/� PX u.x/k1 � ChkC 1

2

X ,�kukN� . (9)

where, the filled distance is defined as on (5). Thus, interpolation with �k provides at least approximation order kC 12 .

Combining Theorems 2.3 and 4.4, we have

kf .x/�LG f .x/k1 D kf .x/�LDf .x/�PX E.x/k1D k.I �PX /.f .x/�LDf .x//k1

� kI �PX /kopkf .x/�LDf .x//k1

� C�

C1h2C C2chC C3c2 log h�

hkC 1

2

X ,�kf 00k1,

where kAkop denotes the operator norm of A.The error estimate of the quasi-interpolation operator LG f is outlined with the following result.

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Theorem 4.5For every f .x/ 2 Hs.R/, and c2j log cj DO.h2/ and hX ,� DO.h/, then there exists constanteC such that

kf .x/�LG f .x/k1 DeCh52Ckkf 00k1.

Remark 4.1Our scheme LG f will encounter the ill-conditioned problem inevitably if we want to achieve higher approximation order. Thus, it relieson the position or filled distance of these scattered data. So far, we only know that it will obtain higher approximation order if the datais relatively equally distributed from numerical experiments. For general case, it is hard to give the theoretical analysis.

5. Generalize to higher dimension

In this section, we extend the univariate quasi-interpolation formula LG f to two dimension on rectangular grid. The extension to 3D orhigher dimension is straightforward.

Given data .xi , yj , f .xi , yj//, iD 0, 1, � � � , n, jD 0, 1, � � � , m. We define another MQ basis as

e� i.y/Dq.y � yj/2C d2,

where d is also a shape parameter.Define a set of functions as follows

e 0.y/D1

2Ce�1.y/� .y � y0/

2.y1 � y0/,

e 1.y/De�2.y/�e�1.y/

2.y2 � y1/�e�1.y/� .y � y0/

2.y1 � y0/,

e i.y/De� iC1.y/�e� i.y/

2.yiC1 � yi/�e� i.y/�e� i�1.y/

2.yi � yi�1/, iD 2, � � � , m� 2,

e m�1.y/D.ym � y/�e�m�1.y/

2.ym � ym�1/�e�m�1.y/�e�m�2.y/

2.yn�1 � yn�2/,

e m.y/D1

2Ce�m�1.y/� .ym � y/

2.ym � ym�1/.

Then, the 2D quasi-interpolation formula of tensor type is given by

eLDf .x/DnX

iD0

mXjD0

f .xi , yj/ i.x/e j.y/. (10)

Similarly, we define an error function in the form

E.x, y/D f .x, y/�eLDf .x, y/.

Suppose it requires to interpolate another set of scattered data X D˚Oxj , Oyj ,

�NjD0 and the corresponding function values

˚f�Oxj , Oyj

��NjD0,

where n is far less than n�m, we have a RBF interpolation function PX E.x, y/ on the set X and denoted by

PX E.x, y/DNX

iD0

ci�.r/, rDq�

x � Oxj�2C�

y � Oyj�2/.

The coefficients can be determined according to the following linear system

PX E�Oxj , Oyj

�D E

�Oxj , Oyj

�, jD 0, 1, � � � , N.

Therefore, the MQ quasi-interpolation operator is constructed by

fLG f .x, y/DeLDf .x, y/CPX E.x, y/. (11)

It is trivial to verify the following fact

Theorem 5.1MQ quasi-interpolation operator fLG f satisfies interpolation property on the setX and possess linear polynomial reproducing property.


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6. Applications and numerical experiments

6.1. Univariate function approximation

In this subsection, we use LG f to approximate two univariate functions, and to compare the errors with those existing operators LDf ,LRf and LW 2f . For simplicity, X D fxig

niD0 and X D fxig

NiD0 are chosen to be the equally spaced points on Œ0, 1�, respectively and

X � X . The shape parameter c of MQ basis is set to be c D 1=n, and Wendland’s CSRBF �.r/ in RBF interpolation is chosen to be�.r/ D .1� r/3C.3rC 1/, where we scale the support size of the basis proportional to the filled distance [2, 22]. It is point out that then also denotes the number of points of RBF interpolant Sf 00 ,N to approximate the second derivative of f .x/ in LW2 f , and the number ofsmaller set of data of LDfxk.j/g

f in LRf , respectively (see [9, 13] for detail).

Here, we adopt l1 and l2 error norms that are evaluated at 500 random testing points yi on Œ0, 1�. The error norms are defined as

l1 D max1�i� testnum

jf .yi/�Lf .yi/j,

l2 D

vuuttestnumXiD1

.f .yi/�Lf .yi//2.

Example 6.1We consider the function [9, 13]

f1.x/D sin.4.5x/, x 2 Œ0, 1�.

The graph of f1.x/ is shown in the left of Figure 1. The l1 and l2 errors by using the LDf , LRf , LW 2f , and LG f for f1.x/with differentnumber of n and n on Œ0, 1� are shown in Tables I and II, respectively.

From Example 6.1, we can see that the convergence rate LG f is faster than LDf and LRf . Moreover, the convergence rate LG f isslower than LW2f . However, the interpolation matrix in LW2f is ill-conditioned when n becomes larger. This problem can be avoidedin LG f because we use CSRBF interpolation at a relatively small set of interpolation data, which guarantees the interpolation matrix issparse and numerically stable.

Example 6.2We consider the function [9, 13]

f2.x/D sin.�x/C 0.1 sin.32�x/, x 2 Œ0, 1�.

The graph of f2.x/ is shown in the right of Figure 1. The l1 and l2 errors by using the LDf , LRf , LW2f , and LG f for f2.x/ withdifferent number of n on Œ0, 1� are shown in Tables III and IV, respectively. Here, n is fixed to be 8.

It is noted that the convergence rate of LG f is more rapid than LDf and LRf when the number of approximated sites is larger.Conversely, we found that the approximation error is not very sensitive to the number of n when n is fixed.

0.2 0.4 0.6 0.8 1.0

1.0

0.5

1.0

05

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Figure 1. The graphs of f1.x/ and f2.x/.

Table I. l1 errors for different quasi-interpolation operators for f1.x/.

.n, N/ LDf [8] LRf [9] LW2f [13] LG f

(32,4) 3.02� 10�2 2.89� 10�2 1.99� 10�3 1.21� 10�2

(32,8) 3.33� 10�2 8.31� 10�4 5.21� 10�3

(64,4) 9.27� 10�3 8.97� 10�3 6.71� 10�4 3.34� 10�3

(64,8) 4.11� 10�3 2.06� 10�4 2.01� 10�3

(128,4) 4.72� 10�3 2.11� 10�4 1.01� 10�3

(128,8) 2.74� 10�3 5.72� 10�3 6.89� 10�5 6.99� 10�4

(128,16) 9.58� 10�3 1.78� 10�5 4.38� 10�415

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Table II. l2 errors for different quasi-interpolation operators for f1.x/.

.n, N/ LDf [8] LRf [9] LW2f [13] LG f

(32,4) 4.35� 10�1 4.21� 10�1 2.79� 10�2 9.29� 10�2

(32,8) 4.42� 10�1 1.03� 10�2 3.56� 10�2

(64,4) 1.36� 10�1 1.34� 10�1 9.26� 10�3 3.19� 10�2

(64,8) 1.55� 10�1 3.22� 10�3 1.02� 10�2

(128,4) 4.09� 10�2 2.91� 10�3 9.59� 10�3

(128,8) 3.99� 10�2 3.87� 10�2 9.95� 10�4 3.63� 10�3

(128,16) 4.82� 10�2 4.58� 10�4 1.21� 10�3

Table III. l1 errors for different quasi-interpolation operators for f2.x/.

n LDf [8] LRf [9] LW2f [13] LG f

64 1.56� 10�1 1.21� 10�1 1.01� 10�1 1.11� 10�1

128 4.94� 10�2 4.57� 10�2 4.01� 10�2 4.36� 10�2

256 1.56� 10�2 9.41� 10�3 3.01� 10�3 8.42� 10�3

400 9.74� 10�3 5.21� 10�3 5.67� 10�4 1.51� 10�3

512 3.74� 10�3 8.72� 10�4 5.11� 10�5 2.03� 10�4

Table IV. l2 errors for different quasi-interpolation operators for f2.x/.

n LDf [8] LRf [9] LW2f [13] LG f

64 1.48 1.24 1.01 1.12128 5.41� 10�1 5.36� 10�1 5.14� 10�1 5.27� 10�1

256 2.17� 10�1 2.01� 10�1 1.81� 10�1 2.06� 10�1

400 8.17� 10�2 6.25� 10�2 3.67� 10�2 5.61� 10�2

512 2.39� 10�2 1.77� 10�2 8.17� 10�3 1.09� 10�2

6.2. Multivariate function approximation

Here, we use bivariate quasi-interpolation operators fLG f to approximate Franke’s function

f .x, y/D�3

4

he�1=4.9x�2/2�1=4.9y�2/2

C e�1=49.9xC1/2�1=10.9yC1/2i

C1

2e�1=4.9x�7/2�.9y�3/2

�1

5e�.9x�4/2�.9y�7/2

.

on a grid of equally spaced points in the unit square Œ0, 1�2.Here, mD nD 25, shape parameters cD dD 0.04 and ND 200. We adopt Wendland’s CSRBF �.r/D .1� r/4C.4rC1/ and its support

size ˛ D 1=2. Franke’s function and the resulting surface by using fLG f are simultaneously shown in Figure 2.To evaluate the approximation behavior, we choose 800 arbitrary points as a testing set. The l1 and l2 errors between f .x, y/ andfLG f are 5.13� 10�3 and 3.71� 10�2, respectively.

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

Figure 2. Franke’s function f.x, y/ and the resulting surface by using NLG f .


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0.2 0.4 0.6 0.8

0.5

0.5

1.0

1.5

2.0

0.5 0.5

40

35

30

25

20

Figure 3. Two sets of scattered data: (A) on the left and (B) on the right.

0.2 0.4 0.6 0.8 1.0

0.5

0.5

1.0

1.5

2.0

0.2 0.4 0.6 0.8 1.0

0.5

0.5

1.0

1.5

2.0

Figure 4. Two resulting curves by using LD f and LG f for scattered data (A).

1.0 0.5 0.5 1.0

55

45

40

35

30

25

20

1.0 0.5 0.5 1.0

60

50

40

30

20

Figure 5. Two resulting curves by using LD f and LG f for scattered data (B).

6.3. Scattered data fitting

Given two sets of scattered data (A) and (B) (Figure 3). Two resulting curves by using quasi-interpolation operators LDf and LG f areshown in Figures 4 and 5, respectively. These interpolated data denoted by red dots are also shown in the right of these figures. Here,the interpolated scattered data X in LG f differs from the set X . It is required that X is closer to the set X .

7. Conclusion

In this paper, we construct a new class of MQ quasi-interpolation operator LG f based on Wu-Schaback’s quasi-interpolation operatorLDf and RBF interpolation. The proposed operator has the advantages linear polynomial reproducing property and interpolationproperty. Besides, it has good approximation behavior. From the numerical experiments, it can be successfully applied to functionapproximation and scattered data fitting.

Acknowledgements

We appreciate the reviewers and editors for their careful reading, valuable comments, timely review, and reply. This work was supportedby the National Natural Science Foundation of China (Grant Nos.11101366, 11226329, 11271328, and 61272307), the Zhejiang ProvincialNatural Science Foundation of China (Grant No. LQ13A010004), and the Foundation of Zhejiang Educational Committee(Grant No.Y201222928).

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a new multiquadric quasi-interpolation operator with interpolation property

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