A family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial reproduction

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  • Journal of Computational and Applied Mathematics 274 (2015) 88108

    Contents lists available at ScienceDirect

    Journal of Computational and AppliedMathematics

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

    A family of multivariate multiquadric quasi-interpolationoperators with higher degree polynomial reproduction

    Ruifeng Wu, Tieru Wu, Huilai Li

    School of Mathematics, Jilin University, Changchun 130012, China

    h i g h l i g h t s

    A family of multivariate multiquadric quasi-interpolants is proposed. There is no demand for derivatives of approximated function in quasi-interpolants. The quasi-interpolants satisfy any degree polynomial reproduction property. The quasi-interpolants can reach up to a higher approximation order.

    a r t i c l e i n f o

    Article history:Received 5 August 2013Received in revised form 24 June 2014

    Keywords:Quasi-interpolationMultiquadric functionsDimension-splittingLagrange basis functionsPolynomial reproductionConvergence rate

    a b s t r a c t

    In this paper, by using multivariate divided difference (Rabut, 2001) to approximate thepartial derivative and the idea of the superposition (Waldron, 2009), we modify a multi-quadric quasi-interpolation operator (Ling, 2004) based on adimension-splitting techniquewith the property of linear reproducing to gridded data onmulti-dimensional spaces, suchthat a family of proposedmultivariatemultiquadric quasi-interpolation operatorsr+1 hasthe property of r + 1 (r Z, r > 0) degree polynomial reproducing and converges up toa rate of r + 2. In addition, the proposed quasi-interpolation operator only demands infor-mation of location points rather than the derivatives of the function approximated. More-over, we give the approximation error of our quasi-interpolation operator. Finally, somenumerical experiments are shown to confirm the approximation capacity of our quasi-interpolation operator.

    2014 Elsevier B.V. All rights reserved.

    1. Introduction

    The approximation of themultivariate function from scattered data is an important theme in numerical mathematics. Al-though the classical method to attack the problem is interpolation, the interpolationmatrix quickly becomes ill-conditionedas the number of points increases. There are different ways to overcome this ill-conditioned problem. Compared with in-terpolation, the quasi-interpolation method does not require solution of any linear system, and can reach up to the ex-pected convergence rate. In this paper, we will focus on the quasi-interpolation method. For a set of functional values{f (qj)}nj=1 taken on a set of nodes = {q0, . . . , qn} R

    d and a set of given quasi-interpolation basis functions (q), the

    Corresponding author. Tel.: +86 0431 85166040.E-mail addresses: ruifengwu13@gmail.com (R. Wu), wutr@jlu.edu.cn (T. Wu), lhljlju@hotmail.com (H. Li).

    http://dx.doi.org/10.1016/j.cam.2014.07.0080377-0427/ 2014 Elsevier B.V. All rights reserved.

    http://dx.doi.org/10.1016/j.cam.2014.07.008http://www.elsevier.com/locate/camhttp://www.elsevier.com/locate/camhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.cam.2014.07.008&domain=pdfmailto:ruifengwu13@gmail.commailto:wutr@jlu.edu.cnmailto:lhljlju@hotmail.comhttp://dx.doi.org/10.1016/j.cam.2014.07.008

  • R. Wu et al. / Journal of Computational and Applied Mathematics 274 (2015) 88108 89

    quasi-interpolation (Qf )(q) of a d-variate function f takes the standard form via linear combination

    (Qf )(q) =n

    j=0

    f (qj) (q), q Rd. (1)

    The approximation properties of the quasi-interpolation operator in the case that qj are nodes of a uniform grid can bewell-understood. For example, for a uniform grid with spacing h, the following quasi-interpolation operator

    (Qf )(q) =jZd

    f (jh)qh

    j

    (2)

    proposed by Rabut [1] can be studied via the theory of principal shift-invariant spaces (see, e.g., [2]), which has been devel-oped by de Boor et al. [3]. Here, the quasi-interpolation basis function is supposed to be a compactly supported or rapidlydecaying function. In [4], Bozzini, Dyn, and Rossini presented a procedure in spaces ofm-harmonic splines inRd which startsfrom a simple generator0 and recursively defines generators1,2, . . . , m1 with corresponding quasi-interpolation op-erators defined as (2) reproducing polynomials of degrees 3, 5, . . . , 2m1 respectively. In [5,6], the StrangFix condition isconsidered a necessary and sufficient condition for the polynomial reproduction, convergence and approximation order ofthe quasi-interpolation. Based on the StrangFix condition for , the scattered data quasi-interpolation by functions, whichreproduces polynomials, has been proposed by Buhmann et al. [7], Dyn and Ron [8], Wu and Liu [9], andWu and Xiong [10].Wewill discuss themultivariatemultiquadric quasi-interpolation operator which can reproduce higher degree polynomialsin this paper.

    With the multiquadric function

    i(x) =(x xi)2 + c2

    1/2, x, xi R, c > 0 (3)

    proposed by Hardy [11], Beast and Powell [12] first constructed a univariate quasi-interpolation operator by shifts of themultiquadric basis function of first degree to finite scattered data, which reproduces linear polynomials. However, theoperator requires the derivatives of the function approximated at endpoints, which is not convenient for practical purposes.Thus, Wu and Schaback [13] constructed another univariate multiquadric quasi-interpolation operator with modificationsat endpoints without derivatives.

    Ling [14] extended the univariate quasi-interpolation operator (see, e.g., [13]) to multi-dimension by using thedimension-splitting univariate multiquadric basis function approach, which reproduces linear polynomials. Based on [14],Feng and Zhou [15] proposed a multivariate dimension-splitting multiquadric quasi-interpolation operator satisfyingquadratic polynomial reproduction property, and proved that it converged with a rate of O(h3) as c = O(h), h = max{h1,h2}, 2h1 = max06i6n1{|xi+1 xi|}, 2h2 = max06j6m1{|yj+1 yj|}. In [16], for any given quasi-interpolation operatorwhich reproduces all polynomials of degree 6 m, Waldron proposed a quasi-interpolation operator which reproduces allpolynomials of degree 6 m + r , where r is a non-negative integer. However, it involves the derivatives of the functionapproximated at each knot.

    The aim of our paper is to present a family of multivariate multiquadric quasi-interpolation operators with higher accu-racy for gridded data, which does not require the partial derivatives of the function approximated at each knot. By virtue ofthe constructive idea in [16] and the multivariate divided difference formula in [17], we extend the operator [14] and givea family of multivariate multiquadric quasi-interpolation operators based on the dimension-splitting technique for griddeddata. We show that the new operator could reproduce any degree polynomials and have the relevant order approximationrate under a certain assumption. Our error estimate indicates that the convergence rate depends heavily on the shape pa-rameter c. Thus, our operator could provide the desired smoothness and precision by choosing a suitable value of the shapeparameter c .

    The remainder of this paper is organized as follows. In Section 2,we introduce somenecessary definitions. In Section 3,wepropose a family of multivariate multiquadric quasi-interpolation operators satisfying any degree polynomial reproductionproperty. In Section 4, we obtain the error estimate. In Section 5, we give some numerical experiments, and verify the rela-tionship between the approximation order and the degree of polynomial reproducing. In Section 6, we give the conclusions,and put forward the future work.

    2. Preliminaries

    Definition 1 ([15]). Let k N, c > 0. The multiquadric functions of 2k 1 degree (2k order) is defined by

    (q; 2k) =q2 + c2

    (2k1)/2, q = (x1, . . . , xd) Rd, (4)

    where c is called the shape parameter, Rd is the d-dimensional Euclidean space, is the Euclidean norm, and j,2k(q) is ashift of (q; 2k) centered at qj:

    j,2k(q) = (q qj; 2k), qj = (xj1, . . . , xjd) Rd. (5)

  • 90 R. Wu et al. / Journal of Computational and Applied Mathematics 274 (2015) 88108

    Definition 2 ([15]). For a given point set {qi = (xi1, . . . , xid)}ni=0 and a set of real values {f (qi)}ni=0, is a bounded

    domain on Rd, the quasi-interpolation (Lf )(q) of a function f : Rd R is defined as follows:

    (Lf )(q) =n

    j=0

    f (qj)j(q), q Rd, (6)

    where {j(q)}nj=0 are quasi-interpolation basis functions.

    Definition 3 ([15]). For any real-coefficient polynomial p of degreem, if Lp = p, then the quasi-interpolation operator L issaid to be satisfying the reproduction property of polynomials of degree no more thanm.

    Definition 4 ([17]). Let F = {f |f : Rd R} and let A be a discrete subset of Rd, = (a1, . . . , ad) Nd = Zd+, || =a1 + + ad. Suppose that D f := Da1 Dad f =

    ||fx

    a11 x

    a22 x

    add

    is the order derivative, Pr+1 = Pr+1(Rd) is the set of

    multivariate polynomials of degree 6 r + 1, where r is a non-negative integer. An operator DA : F F is said to be aPr+1-exact A-discretization of D if and only if(i) There exists a real vector = (a)aA such that, for any f F ,

    DA f () =aA

    af ( + a). (7)

    (ii) For any p Pr+1,

    DAp() = Dp(). (8)

    In such situation, we also say that DA f is a Pr+1-exact A-discretization of D f . Suppose that the points in the set A =

    {q0, . . . , qN1} are properly posed for Pr+1(Rd), that is, there does not exist a nonzero polynomial in Pr+1(Rd) which van-ishes on all the points of A, where N =

    r + 1 + dd

    is the dimension of Pr+1(Rd), and Pr+1(Rd) is the multivariate polynomial

    space of all polynomials of degree 6 r + 1. Then, DA f is determined uniquely.For the sake of simplification, we set d = 2, then obtain N =

    r + 1 + 22

    = (r +3)(r +2)/2. Meanwhile, Pr+1 = Pr+1(R2)

    is the bivariate polynomial space of all polynomials of degree 6 r + 1.

    Definition 5 ([17]). The set A is said to be Pr+1-unisolvent if and only if for all vectors y = (ya)aA, there exists a uniquepolynomial p Pr+1 such that for any a A, p(a) = ya.

    In the next section, we give explicit forms of DA f and the associated (a)aA. Since these forms are ratios of determinants,we need to order the points of A and the basis of Pr+1.

    Definition 6 (Order on A [17]). When necessary in what follows, the points of the set A are assumed to be ordered in anarbitrary (but fixed) way; in this case al1 denotes the (l 1)th element of A (l = 1, . . . ,N) and Al1 denotes the set Al1 ={a0, . . . , al1}.

    Definition 7 (Order on N2 [18]). Let = (a1, a2), = (b1, b2) N2, where N is the class of non-negative integers. Then,(i) = if and only if ak = bk for k = 1, 2.(ii) if either (1) || < || or (2) || = || with strictly preceding with respect to the lexicographic order.(iii) whenever either = or .

    Now, we construct the N N matrix

    M(q; A) =

    (q0 q)0 (q0 q)N1

    .... . .

    ...(qN1 q)0 (qN1 q)N1

    , (9)where (0, 0) = 0 = 0 1 N1.

    Example 1. If A = {q0, . . . , q5} for qi = (xi, yi) R2 and if q = (x, y) R2, then

    M(q; A) =

    1 Y0 X0 Y 20 X0Y0 X

    20

    1 Y1 X1 Y 21 X1Y1 X21

    1 Y2 X2 Y 22 X2Y2 X22

    1 Y3 X3 Y 23 X3Y3 X23

    1 Y4 X4 Y 24 X4Y4 X24

    1 Y5 X5 Y 25 X5Y5 X25

    ,

    where Xi = xi x, Yi = yi y (i = 0, . . . , 5), and (Xi, Yi) = Xa1i Y

    a2i for any = (a1, a2) N

    2.

  • R. Wu et al. / Journal of Computational and Applied Mathematics 274 (2015) 88108 91

    Definition 8 ([18]). (i) For all , N2, we write =1 if = 0 otherwise .

    (ii) For any N2,Mi (q; A) denotes the matrixM(q; A) but for the row containing qi A replaced by zeros except at theintersection with column where the entry is 1.

    Thus, for all q R2, qi A and 0 N1 we have

    Mi (q; A) =

    ...

    . . ....

    (qi1 q)0 (qi1 q)N10

    N1

    (qi+1 q)0 (qi+1 q)N1...

    . . ....

    . (10)

    Example 2. If = 2 = (1, 0) and ifM(q; A) is the matrix given by (9), then

    M1 (q; A) =

    1 Y0 X0 Y 20 X0Y0 X20

    0 0 1 0 0 0

    1 Y2 X2 Y 22 X2Y2 X22

    1 Y3 X3 Y 23 X3Y3 X23

    1 Y4 X4 Y 24 X4Y4 X24

    1 Y5 X5 Y 25 X5Y5 X25

    .

    Definition 9 ([18]). M(A)Mi (A)

    designates M(q; A)

    Mi (q; A)

    evaluated at the origin q = 0 = (0, 0) R2, that is to

    say,

    M(A) = M ((0, 0); A) and Mi (A) = Mi ((0, 0); A) . (11)

    Definition 10. Let A = {q0, . . . , qN1} be a Pr+1-unisolvent set, we denote the Lagrange basis function of the degree r + 1on A by lrj (q) as follows:

    lrj (q) =detMrj,q(A)

    detM(A)(j = 0, . . . ,N 1), (12)

    where

    Mrj,q(A) =

    .... . .

    ...

    q0j1 qN1j1

    q0 qN1

    q0j+1 qN1j+1

    .... . .

    ...

    , q = (x, y), N1 = (r + 1, 0) N2. (13)

    Example 3. If j = 2 and r = 1, then

    M12,q(A) =

    1 y0 x0 y20 x0y0 x20

    1 y1 x1 y21 x1y1 x21

    1 y x y2 xy x2

    1 y3 x3 y23 x3y3 x23

    1 y4 x4 y24 x4y4 x24

    1 y5 x5 y25 x5y5 x25

    .

    3. Quasi-interpolation operator with any degree polynomial reproduction

    In this section, we first propose a family of multivariate multiquadric quasi-interpolation operators r+1, then give thepolynomial reproduction property.

  • 92 R. Wu et al. / Journal of Computational and Applied Mathematics 274 (2015) 88108

    In the following, we introduce a well-known univariate quasi-interpolation operator LD.For given data {a = x0 < x1 < xn = b}, Wu and Schaback [13] proposed the following quasi-interpolation function

    by the shifts of the multiquadric function of first degree i(x) =(x xi)2 + c2

    1/2, c > 0 to each knot.

    (LDf )(x) =n

    i=0

    f (xi)i(x), (14)

    where

    0(x) =12

    +1(x) (x x0)

    2(x1 x0),

    1(x) =2(x) 1(x)2(x2 x1)

    1(x) (x x0)

    2(x1 x0),

    i(x) =i+1(x) i(x)2(xi+1 xi)

    i(x) i1(x)2(xi xi1)

    (2 6 i 6 n 2),

    n1(x) =(xn x) n1(x)

    2(xn xn1)

    n1(x) n2(x)2(xn1 xn2)

    ,

    n(x) =12

    +n1(x) (xn x)

    2(xn xn1).

    They proved that (LDf )(x) possessed the constant and linear polynomial reproduction property.Ling [14] extended the univariate quasi-interpolation formula (14) to multi-dimension using the dimension-splitting

    multiquadric basis function technique. For given data {(xi, yj, fij), i = 0, . . . , n, j = 0, . . . ,m}, the form of dimension-splitting quasi-interpolation for the multiquadric basis function is

    (1f )(x, y) =n

    i=0

    mj=0

    fiji(x)j(y), (15)

    where {i(x)}ni=0 are given above. Along that y direction, the basis functions {j(y)}mj=0 are defined as follows

    0(y) =12

    +1(y) (y y0)

    2(y1 y0),

    1(y) =2(y) 1(y)2(y2 y1)

    1(y) (y y0)

    2(y1 y0),

    j(y) =j+1(y) j(y)2(yj+1 yj)

    j(y) j1(y)2(yj yj1)

    (2 6 j 6 m 2),

    m1(y) =(ym y) m1(y)

    2(ym ym1)

    m1(y) m2(y)2(ym1 ym2)

    ,

    m(y) =12

    +m1(y) (ym y)

    2(ym ym1).

    Feng and Zhou [15] proved that 1 satisfied the property of linear reproduction.Based on the above operator1, we give a family of multivariate dimension-splitting quasi-interpolation operators r+1

    for the multiquadric basis function to gridded data as follows by superposition idea in [16]

    (r+1f )(x, y)=

    ni=0

    mj=0

    f (xi, yj)i(x)j(y) +n

    i=0

    mj=0

    rl=1

    rl

    r+1l

    1l!

    ls=0

    ls

    (x xi)ls(y yj)s

    lfxlsys

    (xi, yj)

    i(x)j(y)=

    ni=0

    mj=0

    r

    l=0

    r + 1 lr + 1

    ls=0

    1(l s)!s!

    (x xi)ls(y yj)s lf

    xlsys(xi, yj)

    i(x)j(y). (16)

    The quasi-interpolation operator r+1 has the following polynomial reproduction property.Theorem 1. The quasi-interpolation operator r+1 reproduces all polynomials of degree 6 r + 1.Proof. Let us verify th...

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