Calcolo manuscript No.(will be inserted by the editor)

The bivariate Shepard operator of Bernoulli

type

Teodora Catinas

“Babes-Bolyai” University, Faculty of Mathematics and Computer Science, Str.

Kogalniceanu 1, Cluj-Napoca, Romania, e-mail: tcatinas@math.ubbcluj.ro, web:

www.math.ubbcluj.ro/~ tcatinas.

Received / Revised version:

Abstract. An efficient method for interpolation of very large scat-

tered data sets is the method of Shepard; unfortunately, it has no

good reproduction qualities and it has high computational cost. In

this paper we introduce a new operator which diminishes the men-

tioned drawbacks. This operator results from the combination of the

Shepard operator with a new interpolation operator, recently pro-

posed by F. Costabile and F. Dell’Accio, and generalizes to the two

variate functions the Shepard-Bernoulli operator introduced in [2].

We study this combined operator and give some error bounds in

This work has been supported by MEdC under grant CEEX ET 18,

3233/17.10.2005.

2 Teodora Catinas

terms of the modulus of continuity of high order and of the mesh

length. We improve the accuracy and the computational efficiency

using a method introduced by R. Franke and G. Nielson.

Keywords: Shepard operator, Bernoulli operator, interpolation

of scattered data, error estimations.

MSC 2000 Subject Classification: 41A05, 41A25, 41A80.

1. Preliminaries

The Shepard method, introduced in 1968, is a well suited method

for multivariate interpolation of very large scattered data sets. It has

the advantages of a small storage requirement and an easy gener-

alization to additional independent variables, but it suffers from no

good reproduction quality, (i.e., it reproduces only constant func-

tions), low accuracy and a high computational cost relative to some

alternative methods [20]. In this paper we introduce a combined oper-

ator of Bernoulli type, using the classical Shepard operator and then

using a modified Shepard method, described in [20]. They preserve

the advantages and improve the reproduction qualities, have better

accuracy and better computational efficiency.

The Shepard operator

The bivariate Shepard operator of Bernoulli type 3

Considering the interpolation points (xi, f(xi)), i = 0, ..., N, Shep-

ard introduced in [22] the linear interpolation operator

(Sµf)(x) =N

∑

i=0

Ai,µ(x)f(xi), µ > 0,

were for each i = 0, ..., N the functions

Ai,µ(x) :=‖x − xi‖

−µ

N∑

k=0

‖x − xk‖−µ

satisfy the conditions

Ai,µ(xk) = δik, i, k = 0, ..., N. (1)

It follows from the definition that

N∑

i=0

Ai,µ(x) = 1. (2)

As a consequence, (Sµf)(x) reproduces exactly only the constant

functions. To avoid this, several authors, starting with Shepard him-

self, have suggested to apply (Sµf)(x) not directly to the f(xi), but

to some interpolation operators P [f, xi](x) at xi by considering the

so called combined operator (see, e.g., [2]-[10], [15]):

(SµPf)(x) =

N∑

i=0

Ai,µ(x)P [f, xi](x), µ > 0. (3)

Function (SµPf)(x) still interpolates f at xi, i = 0, ..., N in virtue of

(1). By (2) it follows that the algebraic degree of exactness, dex(SµPf),

of the operator (3) is equal to maxi=0,...,N

dex(P [f, xi]).

4 Teodora Catinas

The bivariate Shepard operator

Let f be a real-valued function defined on X ⊂ R2, and (xi, yi) ∈

X, i = 0, ..., N some distinct points. Denote by ri (x, y) the dis-

tances between a given point (x, y) ∈ X and the points (xi, yi) , i =

0, 1, ..., N .

The Shepard interpolation operator is defined by

(Sµf) (x, y) =

N∑

i=0

Ai,µ (x, y) f (xi, yi) ,

where

Ai,µ (x, y) =

N∏

j=0j 6=i

rµj (x, y)

N∑

k=0

N∏

j=0j 6=k

rµj (x, y)

, (4)

with the parameter µ > 0.

It is known that the bivariate Shepard operator Sµ reproduces

only the constants and that the function Sµf has flat spots in the

neighborhood of all data points.

The Bernoulli polynomial expansion over a rectangle

We recall some results from [11] and [12]. The Bernoulli polyno-

mials are defined by

B0(x) = 1,

B′n(x) = nBn−1(x), n ≥ 1,

∫ 1

0Bn(x)dx = 0, n ≥ 1.

The bivariate Shepard operator of Bernoulli type 5

The values of Bn(x) at x = 0 are the Bernoulli numbers and they are

denoted by Bn.

For a given function f ∈ Cm[a, b], m ≥ 1, we have the univariate

Bernoulli interpolation formula, (see, e.g., [12]):

f(x) = (Bmf)(x) + (Rmf)(x),

with

(Bmf)(x) := Bm[f ; a, b] = f(a) +m

∑

i=1

Si

(

x−ah

)

hi−1

i! ∆hf (i−1)(a), (5)

where h = b − a and

Si

(

x−ah

)

= Bi

(

x−ah

)

− Bi, (6)

∆hf (i−1)(a) = f (i−1)(b) − f (i−1)(a), 1 ≤ i ≤ m,

and Rmf denotes the remainder term.

Let X = [a, b] × [c, d] be a rectangular domain in R2. We denote

by Cm,n(X) the space of functions f : X → R such that there exist

the continuous partial derivatives

f (i,j)(x, y) = ∂i+j

∂xi∂yj f(x, y), ∀(x, y) ∈ X,

for all (i, j), 0 ≤ i ≤ m, 0 ≤ j ≤ n.

6 Teodora Catinas

Let us denote h := b − a, k := d − c and introduce the following

operators [12]:

∆(h,0)f(x, y) = f(x + h, y) − f(x, y),

∆(0,k)f(x, y) = f(x, y + k) − f(x, y),

∆(h,k)f(x, y) = ∆(h,0)∆(0,k)f(x, y)

= ∆(0,k)∆(h,0)f(x, y)

= f(x, y) − f(x + h, y) + f(x + h, y + k) − f(x, y + k).

According to [12], for a given function f ∈ C(m,n)(X), the inter-

polant of Bernoulli type is the polynomial of degree (m,n) of the

variables x, y, given by

(Bm,nf)(x, y) :=Bm,n[f ; (a, c); (h, k)] (7)

=f(a, c) +

m∑

i=1

∆(h,0)f(i−1,0)(a, c)hi−1

i! Si

(

x−ah

)

+

n∑

j=1

∆(0,k)f(0,j−1)(a, c)kj−1

j! Sj

(

y−ck

)

+m

∑

i=1

n∑

j=1

∆(h,k)f(i−1,j−1)(a, c)hi−1kj−1

i!j! Si

(

x−ah

)

Sj

(

y−ck

)

,

where Sk, k > 1 are given in (6).

The bivariate Shepard operator of Bernoulli type 7

The polynomial from (7) satisfies the following interpolation con-

ditions [12]:

(Bm,nf)(a, c) = f(a, c), (8)

(∆(h,0)Bm,nf)(i,0)(a, c) = ∆(h,0)f(i,0)(a, c), 0 ≤ i ≤ m − 1,

(∆(0,k)Bm,nf)(0,j)(a, c) = ∆(0,k)f(0,j)(a, c), 0 ≤ j ≤ n − 1,

(∆(h,k)Bm,nf)(i,j)(a, c) = ∆(h,k)f(i,j)(a, c), 0 ≤ i ≤ m − 1, 0 ≤ j ≤ n − 1.

Proposition 1. [12] The operator Bm,n has the degree of exactness

(m,n).

We have the interpolation formula [12]:

f = Bm,nf + Rm,nf,

where Rm,nf is the remainder term.

For a function f ∈ Cm,n(X) it is easy to prove that

(Bm,nf)(x, y) = (BmBnf)(x, y) = (BnBmf)(x, y),

where Bmf := Bm[f ; a, b] and Bnf := Bn[f ; c, d] are of the form (5).

Further, it follows

f−Bm,nf = (f−Bmf)+[Bm(f−Bnf)−(f−Bnf)]+(f−Bnf). (9)

2. The bivariate Shepard-Bernoulli operator

We introduce an improved Shepard method by combining the Shep-

ard operator with the Bernoulli bivariate operator. We use the Bernoulli

8 Teodora Catinas

bivariate interpolation operator, that is the tensorial extension of a

two-point univariate polynomial interpolant introduced by F. Costa-

bile in [11].

We obtain an extension of the univariate version of the same com-

bination, introduced in [2].

For a function f ∈ C(m,n)(X), X = [a, b] × [c, d] and a fixed set

of N + 1 distinct points (xi, yi) ∈ X, i = 0, ..., N, we introduce the

bivariate Shepard-Bernoulli operator SB as follows:

(SBf)(x, y) =

N∑

i=0

Ai,µ(x, y)Bim,n[f ; (xi, yi), (hi, ki)], (10)

where Ai,µ, i = 0, ..., N, are defined by

Ai,µ (x, y) =

N∏

j=0j 6=i

rµj (x, y)

N∑

k=0

N∏

j=0j 6=k

rµj (x, y)

, (11)

for a given parameter µ > 0. The polynomials (Bim,nf)(x, y) :=

Bm,n[f ; (xi, yi), (hi, ki)], i = 0, ..., N, denote the Bernoulli bivariate

interpolant in the rectangle with opposite vertices (xi, yi), (xi+1, yi+1)

and they are given by (7), having hi = xi+1 − xi, ki = yi+1 − yi,

i = 0, ..., N , considering a fictive node (xN+1, yN+1) = (xN−1, yN−1).

Remark 1. We can use the fictive node (xN+1, yN+1) = (xN−1, yN−1)

since

Bm,n[f ; (xN , yN ), (hN , kN )] ≡ Bm,n[f ; (xN−1, yN−1), (hN−1, kN−1)],

The bivariate Shepard operator of Bernoulli type 9

as it is noticed in [2].

By (11) it follows that

N∑

i=0

Ai,µ (x, y) = 1. (12)

Theorem 1. For f ∈ C(m,n)(X) the operator SB has the following

interpolation properties:

(SBf)(xp, yp) = f(xp, yp),

for 0 ≤ p ≤ N and µ > m + n − 2.

Proof. It is not difficult to show the following relations:

A(p,q)k,µ (xi, yi) = 0, 0 ≤ i ≤ N, i 6= k

A(p,q)k,µ (xk, yk) = 0,

considering p + q ≥ 1 and for all k = 0, ..., N, 0 ≤ p ≤ m− 1, 0 ≤ q ≤

n − 1 and µ > max{p + q | 0 ≤ p ≤ m − 1, 0 ≤ q ≤ n − 1}. Taking

into account these relations, the cardinality property of Ai,µ’s and

the interpolation properties (8) of Bim,nf, i = 0, ..., N, the conclusion

follows.

Theorem 2. The degree of exactness of the combined operator SB is

(m,n).

10 Teodora Catinas

Proof. By Proposition 1 and relation (12) we get

(SBepq)(x, y) =N

∑

i=0

Ai,µ(x, y)(Bim,nepq)(x, y)

=

N∑

i=0

Ai,µ(x, y)epq(x, y)

= epq(x, y)

N∑

i=0

Ai,µ(x, y) = epq(x, y),

for each polynomial epq(x, y) of degree (p, q), with 0 ≤ p ≤ m, 0 ≤

q ≤ n. Therefore the result is proved.

We obtain the bivariate Shepard-Bernoulli interpolation formula,

f = SBf + RBf,

where SBf is given by (10) and RBf denotes the remainder term.

2.1. Error estimation

Next we give some estimations of the remainder RBf = f − SBf.

We give first an estimation of the interpolation error in terms of the

mesh length and using the modulus of smoothness of order k.

We will use some results given in [13]. Let consider some functions

f ∈ Cm[a, b] and g ∈ Cn[c, d], and the corresponding Bernoulli poly-

nomials Bmf and Bng, for m,n ≥ 1, given by (5). Since the operators

Bm and Bn are quasi-interpolants of order m + 1, n + 1 respectively

The bivariate Shepard operator of Bernoulli type 11

(for the definitions, see for example [13, pp. 134-135]), from Theorem

7.3., p. 225 of [13] it follows the following estimates:

‖f − Bm(f)‖∞ ≤ Cm+1ωm+1 (f ; δ1)∞ , (13)

‖g − Bn(g)‖∞ ≤ Cn+1ωn (g; δ2)∞ ,

where ωk(f ; t)∞ denotes the usual k-th modulus of smoothness of a

function f (see, e.g., [14]), while

δ1 = max0≤i≤N

|xi+1 − xi|, (14)

δ2 = max0≤j≤N

|yj+1 − yj|,

and Cm+1, Cn+1 are some constants.

For a function f : [a, b] → R we consider the univariate combined

Shepard-Bernoulli operator S(1)B , introduced in [2]:

(S(1)B f)(x) =

N∑

i=0

Ai,µ(x)Bm[f ;xi, xi+1],

with Bim := Bm[f ;xi, xi+1] given by (5) and

Ai,µ(x) =

N∏

j=0j 6=i

|x−xj |µ

N∑

k=0

N∏

j=0j 6=k

|x−xj |µ

.

We obtain an estimation of the error for interpolation by S(1)B f in

terms of the modulus of smoothness of high order:

12 Teodora Catinas

Theorem 3. If f ∈ Cm[a, b] then

∥

∥

∥f − S

(1)B f

∥

∥

∥

∞≤ Cm+1ωm+1 (f ; δ1)∞ , (15)

with

δ1 = max0≤j≤N

|xi+1 − xi|.

Proof. We have

(f − S(1)B f)(x) = f(x) −

N∑

i=0

Ai,µ(x)(Bimf)(x)

=

N∑

i=0

Ai,µ(x)f(x) −

N∑

i=0

Ai,µ(x)(Bimf)(x)

=

N∑

i=0

Ai,µ(x)[f(x) − (Bimf)(x)],

and taking into account that∑N

i=0Ai,µ(x) = 1 and relation (13), the

conclusion follows.

Further, we obtain an estimation of the remainder for the bivariate

Shepard-Bernoulli formula, in terms of the modulus of smoothness of

high order.

Theorem 4. If f ∈ Cm,n(X) then

‖RBf‖∞ ≤Cm+1 maxy∈[c,d]

ωm+1 (f(·, y); δ1)∞

+ Cm+1 maxy∈[c,d]

ωm+1 ((f − Bnf)(·, y); δ1)∞

+ Cn+1 maxx∈[a,b]

ωn+1 (f(x, ·); δ2)∞ ,

where δ1 and δ2 are given in (14) and Cm+1, Cn+1 are constants.

The bivariate Shepard operator of Bernoulli type 13

Proof. Taking into account (12) we get

(RBf)(x, y) = f(x, y) −

N∑

i=0

Ai,µ(x, y)(Bim,nf)(x, y)

=N

∑

i=0

Ai,µ(x, y)[f(x, y) − (Bim,nf)(x, y)].

Next, applying relations (9) and (12) it follows that

(RBf)(x, y) =

N∑

i=0

Ai,µ(x, y)(f − Bimf)(x, y)

+N

∑

i=0

Ai,µ(x, y)[Bim(f − Bi

nf)(x, y) − (f − Binf)(x, y)]

+

N∑

i=0

Ai,µ(x, y)(f − Binf)(x, y)

=

[

f(x, y) −N

∑

i=0

Ai,µ(x, y)(Bimf)(x, y)

]

−

N∑

i=0

Ai,µ(x, y)[(f − Binf)(x, y) − Bi

m(f − Binf)(x, y)]

+

[

f(x, y) −N

∑

i=0

Ai,µ(x, y)(Binf)(x, y)

]

.

Applying formula (13) and two times Theorem 3 the conclusion fol-

lows.

14 Teodora Catinas

Theorem 5. For f ∈ Cm,n(X) we have the following form of the

remainder:

(RBf)(x, y) =∑

j<n

∫ b

a

Km+n−j,j(x, y, s)f (m+n−j,j)(s, c)ds (16)

+∑

i<m

∫ d

c

Ki,m+n−i(x, y, t)f (i,m+n−i)(a, t)dt

+

∫∫

X

Km,n(x, y, s, t)f (m,n)(s, t)dsdt,

where Km+n−j,j(x, y, s), Ki,m+n−i(x, y, t), Km,n(x, y, s, t) are the Peano

kernels.

Proof. By Theorem 2 we have that dex(SB) = (m,n) so we can apply

Peano’s Theorem for bidimensional case (see, e.g., [21]). Therefore, it

follows (16) with

Km+n−j,j(x, y, s) = RB

(

(x−s)m+n−j−1

+

(m+n−j−1)!(y−c)j

j!

)

,

Ki,m+n−i(x, y, t) = RB

(

(x−a)i

i!

(y−t)m+n−i−1

+

(m+n−i−1)!

)

,

Km,n(x, y, s, t) = RB

(

(x−s)m−1+

(m−1)!

(y−t)n−1+

(n−1)!

)

,

where

z+ =

z, for z > 0

0, otherwise.

Furthermore,

Km+n−j,j(x, y, s) =(x−s)m+n−j−1

+

(m+n−j−1)!(y−c)j

j!

−N

∑

i=0

Ai,µ(x, y)Bm,n[(x−s)m+n−j−1

+

(m+n−j−1)!(y−c)j

j! ; (xi, yi), (hi, ki)]

The bivariate Shepard operator of Bernoulli type 15

with

Bm,n[(x−s)m+n−j−1

+

(m+n−j−1)!(y−c)j

j! ; (xi, yi), (hi, ki)] =

=(xi−s)m+n−j−1

+

(m+n−j−1)!(yi−c)j

j!

+

m∑

p=1

∆(hi,0)

(

(x−s)m+n−j−1

+

(m+n−j−1)!(yi−c)j

j!

)(p−1,0)∣

∣

∣

∣

∣

x=xi

hp−1

i

p! Sp

(

x−xi

hi

)

+n

∑

q=1

∆(0,ki)

(

(xi−s)m+n−j−1

+

(m+n−j−1)!(y−c)j

j!

)(0,q−1)∣

∣

∣

∣

∣

y=yi

kq−1

i

q! Sq

(

y−yi

ki

)

+m

∑

p=1

n∑

q=1

∆(hi,ki)

(

(x−s)m+n−j−1

+

(m+n−j−1)!(y−c)j

j!

)(p−1,q−1)∣

∣

∣

∣

∣

x=xi,y=yi

·h

p−1

i kq−1

i

p!q! Sp

(

x−xi

hi

)

Sq

(

y−yi

ki

)

,

having

∆(hi,0)

(

(x−s)m+n−j−1

+

(m+n−j−1)!(yi−c)j

j!

)(p−1,0)∣

∣

∣

∣

∣

x=xi

=

= (yi−c)j

j!(m+n−j−p)!

(

(xi + hi − s)m+n−j−p+ − (xi − s)m+n−j−p

+

)

,

∆(0,ki)

(

(xi−s)m+n−j−1

+

(m+n−j−1)!(y−c)j

j!

)(0,q−1)∣

∣

∣

∣

∣

y=yi

=

=(xi−s)m+n−j−1

+

(m+n−j−1)!(j−q+1)!hij−q+1,

and

∆(hi,ki)

(

(x−s)m+n−j−1

+

(m+n−j−1)!(y−c)j

j!

)(p−1,q−1)∣

∣

∣

∣

∣

x=xi,y=yi

=

f(x, y) − f(x + h, y) + f(x + h, y + k) − f(x, y + k)

= (yi−c)j

j!(m+n−j−p)!

(

(xi + hi − s)m+n−j−p+ − (xi − s)m+n−j−p

+

)

.

16 Teodora Catinas

The kernels Ki,m+n−i(x, y, t) and Km,n(x, y, s, t) are calculated in the

same manner.

3. A modified Shepard-Bernoulli operator

In what follows we give a method for improving the accuracy in repro-

ducing a surface with the bivariate Shepard-Bernoulli interpolation,

based on the method introduced by Franke and Nielson in [17] and

improved in [16], [19], [20]. Shepard method has the property that

there are flat spots at each data point and the accuracy tends to de-

crease in the areas where the interpolation nodes are sparse. This can

be improved using the following form of the Shepard operator [20]:

(Sf) (x, y) =

N∑

i=0Wi (x, y) f (xi, yi)

N∑

i=0Wi (x, y)

,

with

Wi (x, y) =[

(Rw−ri)+Rwri

]2,

where Rw is a radius of influence about the node (xi, yi) and it is

varying with i. This is taken as the distance from node i to the jth

closest node to (xi, yi) for j > Nw (Nw is a fixed value) and j as

small as possible within the constraint that the jth closest node is

significantly more distant than the (j − 1)st closest node (see, e.g.

[20]). As it is mentioned in [18], this modified Shepard method is one

The bivariate Shepard operator of Bernoulli type 17

of the most powerful software tools for the multivariate interpolation

of large scattered data sets.

With these assumptions, for f ∈ C(m,n)(X) and distinct points

(xi, yi) ∈ X, i = 0, ..., N, the Shepard-Bernoulli operator, given in

(10), becomes:

(SwBf) (x, y) :=

N∑

i=0

Wi (x, y)Bim,n[f ; (xi, yi), (hi, ki)],

with hi = xi+1 − xi, ki = yi+1 − yi, i = 0, ..., N.

4. Test results

We consider the following test functions ([19], [20], [16]):

Gentle

f1(x, y) = exp[−8116((x − 0.5)2 + (y − 0.5)2)]/3,

Sphere

f2(x, y) =√

64 − 81((x − 0.5)2 + (y − 0.5)2)/9 − 0.5.

The following table contains the maximum and the root mean

squares errors for the Shepard, Shepard-Bernoulli and the modified

Shepard-Bernoulli methods, considering 10 random generated nodes

in the unit square, m = n = 2 and µ = 2. By numerical experiments

we have obtained that for these data the optimal value for Nw is Nw =

8. We compare the obtained numerical results with some combined

18 Teodora Catinas

Shepard operators known in the literature, namely with the combined

Shepard operators of Lagrange, Hermite and Taylor type, denoted

respectively by SL, SH and ST .

From Table 1, one can observe that the error for the Shepard-

Bernoulli interpolation is less than that for Shepard interpolation and

for the combined Shepard interpolation methods considered here, and

the smallest error is for the modified Shepard-Bernoulli.

f1 f2

max error mean squares error max error mean squares error

Sf 0.187044 1.593190 0.237491 1.501829

SBf 0.090581 0.717361 0.027403 0.170273

SwBf 0.062844 0.362312 0.018786 0.087839

SLf 0.535318 2.593516 0.579882 2.276001

SHf 0.464669 1.167781 0.088323 0.194073

ST f 0.211046 0.939458 0.363581 1.297399

Table 1. The interpolation error.

In Figures 1-2 we plot the graphics of fi, SBfi, SwBfi, i = 1, 2.

The bivariate Shepard operator of Bernoulli type 19

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and Sons, New York, 1989

2. Caira, R., Dell’Accio, F.: Shepard-Bernoulli operators. Math. Comp. 76, 299-

321 (2007)

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The bivariate Shepard operator of Bernoulli type 21

Function f1. Interpolant SBf1. Interpolant SwBf1.

Figure 1.

Function f2. Interpolant SBf2. Interpolant SwBf2.

Figure 2.