Abstract
By using the polynomial expansion in the even order Bernoulli polynomials and using the linear combinations of the shifts of the function to approximate the derivatives of , we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants which do not require the derivatives of the function approximated at each node and can satisfy any degree polynomial reproduction property. Error estimate indicates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter and a nonnegative integer . Numerical comparisons show that this technique provides a higher degree of accuracy. Finally, applying our operators to the fitting of discrete solutions of initial value problems, we find that our method has smaller errors than the Runge-Kutta method of order 4 and Wang et al.’s quasi-interpolation scheme.
1. Introduction
Let be a set of distinct points of and let be a function defined on a domain containing . The standard formula for interpolating the function , where has the following form: s.t. for all , where is an interpolation kernel. Many investigators use radial basis functions to solve the interpolation problems (2)-(3). In particular, the multiquadrics presented by Hardy [1], are of especial interest because of their special convergence property, see [2, 3]. Throughout this paper, let the notations and denote the multiquadrics and their shape-preserving parameter as in (4), respectively. A review by Franke [4] indicated that the multiquadric interpolation is one of the best schemes among some 29 interpolation methods in terms of accuracy, efficiency, and easy implementation. Although the multiquadric interpolation is always solvable when the scattered points are distinct [5], the resulting matrix in (2)-(3) quickly becomes ill-conditioned as the number of the scattered points increases. In this paper, we will use the quasi-interpolation technique to overcome the ill-conditioning problem.
A weaker form of (3), well-known as quasi-interpolation, holds for all polynomials of degree no more than , where is nonnegative integer; that is, where . Beatson and Powell [6] first proposed a univariate quasi-interpolation operator which reproduces constants, where in (2) is a linear combination of the multiquadrics, see (46)-(47). Afterwards, Wu and Schaback [7] proposed another quasi-interpolation operator which possesses shape-preserving and linear reproducing properties. They showed that the error of the operator is , when the shape parameter , where . Using the operator , Ling [8] constructed a multilevel quasi-interpolation operator and proved that its convergence order is when . Using the shifts of the cubic multiquadrics, Feng and Li [9] constructed a shape-preserving quasi-interpolation operator. They showed that the operator reproduces all polynomials of degree 2 or less and proved that the convergence rate is as . Combining the operator with Hermite interpolation polynomials, Wang et al. [10] proposed a kind of improved quasi-interpolation operators which reproduce all polynomials of degree . They proved that it converges with a rate of at most. However, the operators require values of the derivatives at endpoints, which are not convenient for practical purposes. Further, many authors offered some examples using multiquadric quasi-interpolation operator to solve differential equations, see [11–15] for details.
Based on CAIRA-DELL’ACCIO’s idea [16], we first define a family of even order Bernoulli-type multiquadric quasi-interpolants by combining the multiquadric quasi-interpolation operator in [6] with the polynomial expansion in even order Bernoulli polynomials in [17]. For practical purposes, applying the divided difference formula in [18] to the operators , we construct a family of modified even order Bernoulli-type multiquadric quasi-interpolants which do not require values of the derivatives at nodes. We prove that the operators reproduce all polynomials of degree and have the convergence rate of under a suitable assumption on the shape parameter . Therefore, our operators can provide the desired smoothness and precision in the practical applications.
The organization of the remainder of this paper is as follows. In Section 2, we briefly recall the definition of Bernoulli polynomials and even order Bernoulli polynomials giving some useful properties. We also obtain three useful theorems for the error in the even order Bernoulli polynomials expansion. In Section 3, we apply previous results to derive a family of modified even order Bernoulli-type multiquadric quasi-interpolants and get their convergence rate. In Section 4, numerical examples are shown to compare the approximation capacity of our new operators with that of CAIRA-DELL’ACCIO’s interpolants and Wang et al.’s quasi-interpolants. In Section 5, we apply our operators to the fitting of discrete solutions of initial value problems for ordinary differential equations. In Section 6, we give the conclusions.
2. Some Remarks on the Polynomial Expansion
2.1. The Generalized Taylor Polynomial
The generalized Taylor polynomial is an expansion in Bernoulli polynomials , that is, the polynomials of the sequence defined recursively by means of the following relations, see [19]: Let function be in the class ; then where the polynomial approximation term is considered as follows: and the remainder term is defined by where denotes the integer part of the argument and . The polynomial approximant has the following result: where is the th Taylor polynomial of with initial point in . According to (10), we denote by the generalized Taylor polynomial.
2.2. The Polynomial Expansion in Even Order Bernoulli Polynomials
Let us consider the polynomial sequence defined recursively by the following relations, see [17]: By (11), the polynomial sequence is related to the following Bernoulli polynomials of even degree, see [17]: We denote by the even order Bernoulli polynomials. For any function in the class , this expansion is realized by the following: where the polynomial expansion in even order Bernoulli polynomials is defined by and the remainder in its Peano's representation is given by where In order to get bounds for remainder (15) even in points outside the interval , we consider the operator where with and . By applying Peano's kernel theorem [20], we give an integral expression for the remainder (15) as follows.
Theorem 1. If and , then for the remainder we have the following integral representations: where and denotes the positive part of the th power of the argument; that is,
Proof. On one hand, there are evaluations of derivatives of up to the order on points and of in the approximation term (14); on the other hand, the exactness of (14) on the set denotes the exactness of the operator on the subset . Applying Peano's kernel theorem, we then obtain where (20) is given by applying the linear functional to a function in . Let ; then Let ; then where is considered a polynomial in of degree . By the expression of , (20) is equal to zero in the interval . Thus, we prove the first case of (19). The remaining cases of (19) can be got in an analogous manner.
By Theorem 1, we can get the following result.
Theorem 2. If and , then for the remainder (18) we get where denotes the sup-norm on and
Proof. Let ; then we find from (19) that Let ; then so that In [21], we have the following known identity: By the identities [17], using relations (30), we get In [17], we have Therefore, by applying (31), we obtain the following form from (29): Further, by applying the third case of (32) and the identities (33), we have Note that the integrands are of type with a that does not change sign in . By applying the first mean value theorem for integrals to (34), we find for some , , , , that After some calculations in (35), we obtain Let ; then we have By the first mean value theorem for integrals, we can get after some calculations where , . By applying relations (36) and (38) to (27), we have By identities (32), we obtain By using (40) in (39), we have after some simplifications that Because we obtain the first case of expression (25). Similarly, we can prove the remaining cases.
Since the polynomials of degree are not greater than , we can obtain the desired bounds in an analogous manner.
Theorem 3. If and , then for the remainder (18) we get where denotes the sup-norm on and
3. The Modified Even Order Bernoulli-Type Quasi-Interpolants
The multiquadric quasi-interpolant [6] is defined by the following: where for , where is the hat function that has the nodes , that is identically zero outside the interval and that satisfies the normalization condition . The operator reproduces constants. Based on the operator , we first define a family of even order Bernoulli-type multiquadric quasi-interpolants as follows: where is the natural extension of the polynomial expansion defined in (14) and . The operators possess the polynomial reproduction property as follows.
Theorem 4. The operators reproduce all univariate polynomials of degree no more than .
Proof. The argument follows from the well-known property since for , where .
Although the quasi-interpolants reproduce all polynomials of degree , they require the derivative of at every node, which are very difficult to measure in practice. Therefore, we use divided difference operator in following Definition 5 to approximate in the operators and then get a family of modified even order Bernoulli-type multiquadric quasi-interpolants .
Definition 5 (see [18]). Let and let be a discrete subset of , . Suppose that is the order derivative. An operator is said to be a -exact -discretization of if and only if(i)there exists a real vector s.t. for any ,
(ii)for any ,
In such situation, we also say that is a -exact -discretization of . Let the points be distinct in the set ; then is determined uniquely.
Let denote the number of elements in set. Let the points in set be distinct and ; then by Definition 5 and [18], a -exact -discretization of the order derivative is
where
By virtue of the location of each pair , , we choose suitable sets and then replace and with and , respectively. Thus, the modification quasi-interpolants can be expressed as follows:
Note that the expressions of in the modification operators are provided by the following theorem.
Theorem 6. For any and , let be a -unisolvent set and denote the set of points of the form , where . Let , , and . Then, for each , we have where
Proof. For each , we set , , and . According to (50) and (53), we get where Therefore, we have Let us set ; then we get the proof of the Theorem 6.
Remark 7. For , we give the expression of the modification operator as follows: where
3.1. The Polynomial Reproduction Properties of the Operators
Theorem 8. The operators reproduce all univariate polynomials of degree no more than .
Proof. By using the proof of Theorem 4 and formulas (51)–(54), we get the proof of Theorem 8 immediately.
3.2. The Convergence Rate of the Operators
In order to obtain the convergence rate of the modified multiquadric quasi-interpolants , we make use of the following notations: where denotes the cardinality function. So, and denotes the maximum number of points from contained in an interval . At first, for the quasi-interpolants , we then give the error estimates as follows.
Theorem 9. Let satisfy where is a positive constant and is a positive integer. Let ; then where and is a positive constant independent of , , and .
Proof. Let each pair , , be fixed and let . For each we make use of the following settings:
By applying (14) to (48), we obtain
where
Assume that
where the set denotes the covering of with half open intervals. Thus, for every , there exists a unique s.t. . Then, we get the following inequalities:
where and . Therefore, we have from (70)
We also obtain from the definition of
On the other hand, when , we get, after some calculations, by applying the first mean value theorem for integrals to (46),
where . When , we obtain in an analogous manner
When , we also obtain
where . Then, for (68), we have
where the last inequality follows from
Let ; then .
Let ; then .
Applying Theorem 3, we can obtain the desired error estimates of the operator in an analogous manner.
Theorem 10. Let satisfy where is a positive constant and is a positive integer. Let ; then where and is a positive constant independent of , , and .
Because of disadvantage with the derivatives in the operators , we give the following desired error estimates of the modification quasi-interpolants .
Theorem 11. Let satisfy where is a positive constant and is a positive integer. Let ; then where and is a positive constant independent of , , and .
Proof.
Consider
The second term of the right-hand sides in (84) has been obtained from Theorem 10, so we only need to prove the first term.
We denote by and the maximum and the minimum distance between adjacent nodes, respectively. Let and .
Let be a constant; then, according to [18], we get
Therefore, we obtain
Let be constants; then
Finally, applying Theorem 10 and (87) to (84), we get
Let , then .
Let , then .
4. Numerical Examples
We consider the following functions on the interval , which are firstly used in [16]: We apply the interpolation operators , the quasi-interpolation operators , and the quasi-interpolants on the above functions with , where and are defined by [10, 16], respectively.
We use uniform grids of 21 points for the operators , , and in Tables 1, 2, 3, and 4. In order to estimate the errors as accurate as possible, we compute the approximation functions at the points , . Tables 1–4 show the mean and max errors which are computed for different values of the parameters , , and . The numerical results show that our quasi-interpolants have better approximation behavior.
5. An Application of the New Operators
After solving the following initial value problems: by virtue of a discrete method, we often need to master the solution on a set of points that differs from the grid. Here we use our operators to solve the problems. In fact, combinations of our operators with discrete solvers of ODEs provide approximations of the solution of the problems (90) on . An algorithm for constructing these quasi-interpolants is given as follows. The discrete solver produces an approximation of the exact solution at nodes , in . Substituting the exact values mentioned above into the definition of our operators by their respective approximations, we get the proposed quasi-interpolants. We consider the initial value problems as follows.
Problem A
Problem B The exact solutions of Problems A and B are and , respectively. By the Runge-Kutta method of order 4, we obtain the on a uniform grid of 21 nodes in . Calculating the approximative functions at points , , we get the mean and max errors in Table 5. Comparing the approximation capacity of our proposed quasi-interpolants with that of Runge-Kutta scheme of order 4 and Wang et al.'s quasi-interpolation scheme [10] in Table 5, we find that our technique has smaller errors in the Problems A and B.
6. Conclusions
In this paper, we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants which reproduce polynomials of higher degree. There is no demand for the derivatives of function approximated at each node in our operators , so they do not increase the orders of smoothness of the function . Under a certain assumption, we give an expected result on the convergence rate of our operators . The numerical examples show that our operators produce higher degree of accuracy. Furthermore, applying the operators to the fitting of discrete solutions of initial value problems, we find that our operators provide more accurate approximation solver.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work was supported by National Natural Science Foundation of China (Grant nos. 61373003 and 11271041).