Quasi-interpolation based on bivariate quadratic B-splines with multipleknots

C. Dagnino1, P. Lamberti1∗, andS. Remogna1

1 Department of Mathematics, University of Torino, via C. Alberto, 10, 10123 Torino, Italy.

Taking into account the results given in [1, 3, 4] on bivariate spline quasi-interpolation and in [5] on the effects of multipleknots, in this paper we consider local quadratic spline quasi-interpolants on criss-cross triangulations of a rectangular domainΩ, with possible multiple knots insideΩ and/or on the boundary∂Ω. Such splines can be particularly useful in mechanicsand engineering applications, since they can model constrained surfaces with different kind of smoothness inΩ.

1 Introduction

Let Tmn be a criss-cross triangulation of a rectangular domainΩ = [a, b] × [c, d], defined byξ = a = ξ0 < ξ1 < . . . <

ξm < ξm+1 = b andη = c = η0 < η1 < . . . < ηn < ηn+1 = d.Given a non negative integer sequencemξ

imi=1, with m

ξi ≤ 2, for all i, setM = 3 +

∑mi=1 m

ξi and lets = si

M+2i=0 be

a nondecreasing sequence so that: (i)s0 = s1 = s2 = ξ0 = a andsM = sM+1 = sM+2 = ξm+1 = b; (ii) for i = 1, . . . , m,the numberξi occurs exactlymξ

i times ins. Similarly, given a non negative integer sequencemηj

nj=1, with m

ηj ≤ 2, for

all j, setN = 3 +∑n

i=1 mηj and lett = tj

N+2j=0 be a nondecreasing sequence so that: (i)t0 = t1 = t2 = η0 = c and

tN = tN+1 = tN+2 = ηn+1 = d; (ii) for j = 1, . . . , n, the numberηj occurs exactlymηj times int.

On Tmn we define the spline spaceS2(Tmn) of all piecewise quadratic polynomials such that the smoothness on eachmesh segmentx = ξi (y = ηj ) is µ

ξi (µη

j ). We recall knot multiplicitymξi (mη

j ) implies decay of spline smoothness, i.e.

µξi = 2 − m

ξi (µη

j = 2 − mηj ) [5].

For such a space, from [7], we can get that dimS2(Tmn) = 8 − mn + m + n + (2 + n)(M − 3) + (2 + m)(N − 3).Moreover we can generate a collection ofM · N B-splinesBij(i,j)∈Kmn

, with knot partitionss andt, spanningS2(Tmn),with Kmn = (i, j) : 0 ≤ i ≤ M − 1, 0 ≤ j ≤ N − 1. In Fig.1 we propose the graph of one of these B-splines with its localsupport.

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.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................| | | |

−

−

−

−tj+1

tj

tj−1

tj−2

si−2si−1

si si+1

kj+1

kj

kj−1

hi−1 hi+1

Fig. 1 A double knot quadraticC0 B-splineBij and its support.

In S2(Tmn) we can define 2D spline q-i’s of kind

Qf(x, y) =∑

(i,j)∈Kmn

λijfBij(x, y) (1)

whereλijf =∑

k w(ij)k f(x

(i)k , y

(j)k ), with (x

(i)k , y

(j)k ) mesh point lying in the support ofBij , andw

(ij)k ∈ IR, w

(ij)k 6= 0, such

thatQf = f, ∀f ∈ IP`, 0 ≤ ` ≤ 2. We remark that the presence of multiple knots allows to model spline surfaces with

∗ Corresponding author E-mail:paola.lamberti@unito.it, Phone: +39 011 670 2829, Fax: +39 011 670 2878

PAMM · Proc. Appl. Math. Mech. 7, 2020003–2020004 (2007) / DOI 10.1002/pamm.200700015

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

different kind of smoothness inΩ. Moreover in case of simple knots (i.e.mξi = m

ηj = 1, ∀i, j) we obtain the spline q-i’s

presented in [1–4,6].

2 Applications

A Matlab code has been carried out to construct the spline q-i’s (1). Now we report some numerical and graphical results,obtained by running our Matlab procedures, for the choices of Q introduced in [1] (Fig.2 and Table 1).

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

a) −1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

b)

Fig. 2 S1f with simple (a) and double (b) knots atx = 0 andy = 0, m = n = 4, f(x, y) =| x | y if xy > 0, f(x, y) = 0 elsewhere.

simple knots double knotsm = n ‖S1f − f‖∞ ‖S2f − f‖∞ ‖W2f − f‖∞ ‖S1f − f‖∞ ‖S2f − f‖∞ ‖W2f − f‖∞

4 6.25(−2) 5.21(−2) 6.25(−2) 3.33(−16) 3.33(−16) 3.33(−16)

8 1.56(−2) 1.30(−2) 1.56(−2) 4.44(−16) 4.44(−16) 4.44(−16)

16 3.91(−3) 3.26(−3) 3.91(−3) 4.44(−16) 4.44(−16) 4.44(−16)

32 9.77(−4) 8.14(−4) 9.77(−4) 4.44(−16) 4.44(−16) 4.44(−16)

64 2.44(−4) 2.03(−4) 2.44(−4) 4.44(−16) 5.55(−16) 4.44(−16)

Table 1 Comparison between maximum errors, by using simple and double knots and for different kind ofQ. These results are due to thefact thatS1f = f for f(x, y) = xhyk, 0 ≤ h, k ≤ 1.

Moreover, in the parametric setting we can act in the same way. For example, we construct a q-i surfaceσ(u, v) =∑(i,j)∈Kmn

cijBij(u, v), with (u, v) ∈ Ω parametric domain and control pointscij = (xij , yij , zij) ∈ IR3 (Fig.3).

c1,0

c2,0

c1,1

c2,1

c1,2

x

c6,0

c0,0

c3,0

c2,2

c6,1

c0,1

c3,1

c4,0

c5,0

c6,2

c0,2

c3,2

c4,1

c5,1

y

c4,2

c5,2

z

0,0,0 1 2 3 4 5,5,50,0,0

1,1,1

a)

c1,0

c2,0

c1,1

c2,1

x

c1,2

c6,0

c0,0

c3,0

c2,2

c6,1

c0,1

c3,1

c4,0

c5,0

c6,2

c0,2

c3,2

c4,1

c5,1

y

c4,2

c5,2

z

0,0,0 1 2,2 3 4,4,40,0,0

1,1,1

b)

Fig. 3 σ with simple and double knots in the parametric domain:s × t = 0, 0, 0, 1, 2, 3, 4, 5, 5, 5 × 0, 0, 0, 1, 1, 1 (a) ands × t =0, 0, 0, 1, 2, 2, 3, 4, 4, 4 × 0, 0, 0, 1, 1, 1 (b).

References

[1] C. Dagnino and P. Lamberti, in: Curve and Surface Fitting: Avignon 2006, edited by A. Cohen, J. L. Merrien, L. L. Schumaker(Nashboro Press, Brentwood, 2007), p. 101.

[2] C. Dagnino and P. Lamberti, J. Comput. Appl. Math., to appear.[3] C. Dagnino and P. Sablonniere, Prepublication IRMAR 06-06 (2006).[4] P. Sablonniere, in: Modern developments in multivariate approximations, edited by W. Hausmann & al., ISNM145 (Birkhauser

Verlag, Basel, 2003), p. 263.[5] R.-H. Wang and C.-J. Li, J. Comp. Math.22(1), 137 (2004).[6] R.-H. Wang, Multivariate Spline Functions and Their Application (Science Press, Beijing/New York, Kluwer Academic Publishers,

Dordrecht/Boston/London, 2001).[7] X.-S. Wang, Northeastern Math.2, 66 (1986).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Contributed Papers 2020004