A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics

A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics

Ren-Hong Wang; Min Xu

Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P.R.China E-mail: [e-mail]

ABSTRACT

In this paper, a kind of Bernoulli-type operator is proposed by combining a univariate multiquadric quasi-interpolation operator with the generalized Taylor polynomial. With an assumption on the shape-preserving parameter c, the convergence rate of the new operator is derived, which indicates that it could produce the desired precision. Numerical comparisons show that this method offers a higher degree of accuracy. Moreover, the associated algorithm is easily implemented.

Mathematical subject classification: 41A05, 41A35.

Key words: radial basis functions, multiquadrics, quasi-interpolation, Bernoulli polynomials, convergence rate.

1 Introduction

The standard formula for interpolating a function ƒ :[a, b] → on scattered points and data , where

has the form

such that

where φ(·) is an interpolation kernel. Many researchers use radial basis functions (RBFs) to solve the interpolation problem (1.2)–(1.3). In particular, the multiquadrics (MQs) introduced by Hardy [10],

are of special interest, because of their spectral convergence property, see [5, 6]. Throughout the rest of this paper, we use the notations φ_j(·) and c to denote the MQs and their shape-preserving parameter as in (1.4), respectively. A review by Franke [9] showed that the MQ interpolation is considered as one of the best methods among 29 scattered points interpolation schemes based on accuracy, stability, efficiency, memory requirement and easy implementation. Although the MQ interpolation is always solvable when the x_j's are distinct [11], the associated coefficient matrix in (1.2) quickly becomes ill-conditioned as the number of points increases. There are different ways to overcome this problem. In this paper, we will focus on the quasi-interpolation method. A weaker form of (1.3), known as quasi-interpolation, holds only for polynomials of degree m, i.e.,

for all 0

j

N, where Π_m denotes the set of polynomials of degree m. Many researchers have investigated the MQ quasi-interpolation method, see [1, 2, 14, 15] for details.

Based on the idea in [7], we combine the multiquadric quasi-interpolation operator proposed in [3] with the generalized Taylor polynomial proposed in [8] to get a Bernoulli-type quasi-interpolation operator. The new operator could reproduce polynomials of higher degree than the operator . We derive the convergence rate of the operator with a suitable assumption on the shape-preserving parameter c, and find that our operator could achieve a convergence rate of higher order by using a smaller parameter, which makes the associated quasi-interpolant less smooth. So we could use an optimal value of c according to the desired smoothness and precision of the quasi-interpolant.

The remainder of this paper is organized as follows. In Section 2, we introduce some previous results about the generalized Taylor polynomial, which plays an important role in the construction of our operator. In Section 3, we propose the Bernoulli-type quasi-interpolation operator, and study its convergence rate. In Section 4, we give numerical experiments to show that the operator is capable of producing high accuracy. In Section 5, we give the conclusions and future work.

2 The generalized Taylor polynomial

The generalized Taylor polynomial is an expansion in Bernoulli polynomials B_n(x), i.e., the polynomials defined by the following generating function

For any function ƒ in the class C^m[a, b](a < b), this expansion is realized by the equation

where the polynomial approximation term P_m[ƒ ; a, b](x) is given by

and the remainder term R_m[ƒ ;a, b](x) is given by

with h = b – a. The polynomial approximant P_m[ƒ ; a, b](x) has the following properties:

where T_m[ƒ ; a](x) is the mth Taylor polynomial of ƒ about the point a. Due tothe property (2.5), we call P_m[ƒ ; a, b](x) the generalized Taylor polynomial. The following two theorems give bounds for the remainder term R_m[ƒ ; a, b](x), see [7].

Theorem 2.1.Let ƒ ∈ C^m[c, d] and x ∈ [c, d]. For the remainder term, we have

where || · ||_∞denotes the sup-norm on [c, d] and

Theorem 2.2. Let ƒ ∈ C^m+1[c, d] and x ∈ [c, d]. For the remainder term, we have

where C(m) is defined by (2.8).

3 The Bernoulli-type quasi-interpolation operator

The quasi-interpolation operator is defined as follows:

where

and

for i = 1,2, ..., N – 1, where B_i(θ) is the piecewise linear hat function having the knots {x_i-1, x_i, x_i+1}, and satisfying B_i(x_i) = 1. The operator reproduces polynomials of zero degree, i.e.,

By combining the operator with the mth generalized Taylor polynomial, we propose a kind of Bernoulli-type quasi-interpolation operator as follows

with x_N+1= x_N-1. The operator has the following polynomial reproduction property.

Theorem 3.1. The Bernoulli-type operator reproduces all polynomials of degree m, i.e.,

Proof. It is easy to show that

where h_i = x_i+1-x_i. Then, by (2.2) and (2.4) we get

Combining it with (3.5), we have (3.7).

Remark 3.1.If P_m[ƒ ; x_i, x_i+1](x) in (3.6) is replaced by the mth Taylor polynomials T_m[ƒ ; x_i], the corresponding operator also reproduces polynomials of degree m. In this case, however, we need to compute the derivatives of ƒ up to order m, which is one greater than the order of derivatives needed in .

In order to study the convergence rates of the new operator, we introduce the following notations

where X = {x₀, x₁, ...,x_N} and #(·) denotes the cardinality function. Therefore, M is the maximum number of points from X contained in an interval I_r(x). For the convergence rate of the Bernoulli-type quasi-interpolation operator , we have the following theorem.

Theorem 3.2. Assume that c satisfies

where D is a positive constant, and l is a positive integer. If ƒ ∈ C^m[a, b], then

where

and K is a positive constant independent of x and X.

Proof. For each pair x_i, x_i+1, we set

and

d^m[x_i, x_i+1](x) = |d[x_i, x_i+1](x)|^m.

We have

where

Let

Q_ρ(u) = (u − ρ, u + ρ], u ∈ [a, b], ρ > 0,

and

T_j = Q_r(x − 2rj) ∪ Q_r(x + 2rj), j = 0, 1, ∙ ∙ ∙ , n,

where [·] denotes the integer part of the argument. Obviously, the set is a covering of [a, b] with half open intervals. Therefore, for each i ∈ {0,1, ..., N} there exists a unique j ∈ {0,1,..., n} such that x_i ∈ T_j. When j 2, the following inequalities hold:

(2j − 1)r |x − x_i| (2j + 1)r,

(2( j − 1) − 1)r |x − ξ_i| (2( j + 1) + 1)r, for ξ_i∈ [x_i−1, x_i+1],

and

|d[x_i, x_i+1](x)| (2( j + 1) + 1)r.

It follows from the definition of M that

1 #(X ∩ T₀) M,

1 #(X ∩ T_j) 2M, j = 1, 2, ∙ ∙ ∙ , n.

When x₀ ∈ T_j (j 2), we have

where ξ₀ ∈ [x₀, x₁]. Similarly, for x_N ∈ T_j(j 2), we have

ψ_N (x) c²r⁻²(2j − 3)⁻².

When x_i(i = 1,2,..., N-1) belong to T_j (j 2), we have

where ξ_i ∈ [x_i-1, x_i+1].

Then we can get

Case 1: (m = 1).

If l = 1, 2r^m+D²r^2l+m-2= (r | ln r |).

If l > 1, 2r^m+D²r^2l+m-2= (r).

Case 2: (m > 1).

If m 2l – 1, 2r^m+ D²r^2l+m-2= (r^m).

=

By using Theorem 2.2, we can prove the following theorem in an analogous manner.

Theorem 3.3. Assume that c satisfies

where D is a positive constant, and l is a positive integer. If ƒ ∈ C^m+1[a, b], then

where

and K' is a positive constant independent of x and X.

Remark 3.2.From Theorem 3.3, we note that the convergence rate do not always achieve the expected maximum m + 1. If the basis functions are replaced by B-splines (for instance, ψ_j(x) degenerate into linear B-splines when c is chosen to be 0), the convergence rate will achieve m + 1. However, the associated quasi-interpolant is finitely differentiable, which is less smooth than f(x).

4 Numerical experiments

We consider the following functions on the interval [0,1]

These functions were firstly proposed in [7] and result from adapting to the univariate case test functions generally used in the multivariate interpolation of large sets of scattered data [13]. We apply the approximating operators , and on each function ƒ_i with c = r^l. The operator is defined as follows

with , see [7] for details.

We use uniform grids of 17 points for , , grids of 11 points for , , and , and finally grids of 8 points for and . In order to estimate the errors as accurate as possible, we compute the approximating functions at the points (i = 1, ..., 100). Tables 1-6 display mean and max errors for different values of the parameters µ, l and m. In these examples, we find that c = r² is an optimal parameter for the fixed precision. The numerical results show that the approximating power of the Bernoulli-type operator with multiquadrics is much better than that of the Shepard-Bernoulli operator.

5 Conclusions

In this paper, a kind of Bernoulli-type quasi-interpolation operator is proposed by combining a univariate multiquadric quasi-interpolation operator with the generalized Taylor polynomial. A result on the convergence rate of the operator is given. Numerical experiments show that the operator is capable of producing high accuracy. Moreover, the associated algorithm is easily implemented.

In our future work, we plan to use the operator to fit scattered data. For instance, we are applying it to the fitting of discrete solutions of initial value problems of ODEs, and good results are obtained.

Acknowledgments. The work was partly supported by the National Natural Science Foundation of China (Grant Nos. 60533060, 10726068 and 10801024) and the Innovation Foundation of the Key Laboratory of High-Temperature Gasdynamics of Chinese Academy of Sciences, China.

Received: 23/II/09.

Accepted: 09/III/09.

#CAM-62/09.

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Publication Dates

• Publication in this collection
  19 Mar 2010
• Date of issue
  2010

History

• Received
  23 Feb 2009
• Accepted
  09 Mar 2009