Abstract
We mainly present the error analysis for two new cubic spline based methods; one is a lacunary interpolation method and the other is a very simple quasi interpolation method. The new methods are able to reconstruct a function and its first two derivatives from noisy function data. The explicit error bounds for the methods are given and proved. Numerical tests and comparisons are performed. Numerical results verify the efficiency of our methods.
1. Introduction
Cubic spline, as the most commonly used spline in practice, is a fundamental approximation tool [1–11]. Nowadays, it has been widely used in many fields such as numerical analysis, computer aided geometric design, mathematical modeling, and engineering problems. Essentially, cubic spline is a twice differentiable piecewise cubic polynomial defined over a partitioned interval.
Mathematically, cubic spline interpolation is often introduced as follows. Let be a function defined over , let be a set of given function data at the nodes and let be two boundary derivatives. Then, there exists a unique cubic spline satisfying
However, we often meet two troubles in the practical applications of cubic spline interpolation. The first trouble is that we cannot obtain the precise function values in (1). They generally involve some unavoidable measurement noise. The second trouble is that it often lacks the boundary derivatives in (3).
To deal with the troubles, in this paper, we give two new effective cubic spline based methods for reconstructing , , and from the given noisy data where is the measurement noise. The first one is a noisy lacunary interpolation method (Method I) and the second one is a very simple noisy quasi interpolation method (Method II). The error bounds of the methods, which have not been studied before and are important and useful for the users of cubic spline, are mainly studied in this paper.
We organize the remainder of this paper as follows. In Section 2, we present some useful preliminaries; in Section 3, we give the new methods; in Section 4, we present the theoretical results of the errors; in Section 5, we perform some numerical tests to verify the error analysis; finally, we conclude this paper in Section 6.
2. Preliminaries
2.1. Cubic B-Splines
We assume that the nodes in (2) are equidistant because this case is very common in practice. The nodes produce a uniform partition for with mesh size . The dimension of the cubic spline space over is . The corresponding cubic B-splines are given below [4–8]. For , let The other six B-splines , , , , , and are generated by the translation, where They are linearly independent, nonnegative, and locally supported. Moreover, The values of , , and at the nodes are listed in Table 1.
2.2. Approximate Boundary Derivatives
Using two-point numerical differentiation formula, we have Similar results can be obtained by using three-point and five-point numerical differentiation formulae. See Tables 2 and 3, where , , , , , and and and represent the computational truncated errors to and . They arise from the used numerical differentiation formulae and the above-mentioned measurement noise.
3. Two New Methods
3.1. Method I
We study the following noisy lacunary cubic spline interpolation (NLCSI) problem. We hope to find a cubic spline satisfying To make the NLCSI problem uniquely solvable, it requires using two approximate boundary derivatives in Section 2.2. Obviously, there also exists a unique noisy lacunary cubic spline satisfying
Let be the cubic spline determined by (11), where the unknown coefficients can be obtained by solving the linear system followed from Table 1. Furthermore, we can use and to approximate and , respectively.
3.2. Method II
By using the given function data, we can directly get a cubic spline where and . We can also use , , and to approximate , , and , respectively.
The method is very simple and effective method for noisy data because it avoids using approximate boundary derivatives and also avoids solving the linear system (12).
4. Main Results
4.1. Error Analysis for Method I
We denote (12) by .
Lemma 1. is invertible and .
Proof. Add column one to column three and also add column to column , and we get a strictly diagonally dominant matrix Obviously, is invertible and , where We have Hence, we have .
Let be the cubic spline determined by (4), and then we have . Let and we have see Table 4 for the results, where and , , and are defined similarly.
Lemma 2. Consider .
Proof. Consider .
Lemma 3. Consider , , and .
Proof. Because of property (8), we only need to check them over a typical subinterval . By differentiating (6), for a general , we have All of them are locally supported over four adjacent subintervals. (i)By the nonnegativity and partition of unity of cubic B-splines, for , we have (ii)For , we have (iii)For , we have
Lemma 4. Let and be the cubic spline interpolants of determined by (4) and (11), respectively. Then we have
Proof. First of all, for , we have And then using Lemmas 2 and 3, we get these results.
Theorem 5. Let be the noisy lacunary cubic spline interpolant of determined by (11). Then one has
Proof. These results follow from the traditional cubic spline interpolation error theory [1, 3, 7], Lemma 4, and the following triangle inequality
4.2. Error Analysis for Method II
From (13) and Table 1, for , we have It is a surprise to find that and are the same as the well-known central numerical differentiation formulae.
Lemma 6. Let be the noisy cubic spline quasi interpolant of determined by (13). Then for , one has
Proof. By (30), we have The proofs of (31) and (32) are similar, which are omitted.
Theorem 7. Let be the noisy cubic spline quasi interpolant of determined by (13). Then we have
Proof. We first prove (37). is a cubic spline; hence is a piecewise continuous linear function over with respect to the partition . Let be the piecewise linear interpolant to with respect to . For , let and be the restriction of and over . Then we have
For , by (38) and (33), we get
Hence, for all , we have
Moreover, by the piecewise linear polynomial interpolation theory [1, 3, 7], for all , we have
Then (37) follows immediately from (40) and (41).
Next, we prove (36). For , let be the restriction of over . Then for , by (32) and (37), we have
Finally, we prove (35). For every subinterval , , we give
They are very useful in cubic Hermite interpolation, and we also have
Let be the restriction of over ; then it can also be written as
Let
be the cubic Hermite interpolant of over ; then for , by using (31), (32), (44), (45), and (46), we have
By the triangle inequality, we get (35).
5. Numerical Tests and Discussions
5.1. Numerical Tests
In this section, we perform numerical tests by Matlab. The following examples are considered.
In every numerical test, the mesh size and the measurement noise bound are both given. Because the measurement noises are random, we let , where are random numbers and satisfy .
In Tables 5 and 6, Methods I-1, I-2, and I-3 represent Method I with two-point, three-point, and five-point approximate boundary derivatives, respectively. CSM represents the cubic spline method in [11]. , , and are the maximum absolute error of the function, the first order derivative, and the second order derivative, respectively.
5.2. Discussions
Generally, the maximum absolute errors , , and vary if one of and does. If is fixed and decreases, then the maximum absolute errors , , and will decrease. But if is fixed while decreases, the errors will not decrease necessarily; they maybe increase sometimes. See the theoretical results in Theorem 5 and Theorem 7 and the numerical results in Tables 5 and 6.
When and are both fixed in a specific test, it is easy to find that and of Method I-2 and Method I-3 are better than those of Method I-1 and Method II, while of Method I-3 and Method II are better than Method I-1 and Method I-2. See Tables 5 and 6.
It is very reasonable to compare our methods with the cubic spline method (CSM) in [11] because our methods are also based on cubic spline. From Tables 5 and 6, we find that the errors of Method I-3 are overall better than CSM in [11]. At the same time, of Method II are better than CSM in [11]. In summary, when approximating a function, we advise using Method I-3, Method I-2, and CSM [11]; when approximating its first order derivative, we advise using Method I-3 and CSM [11]; when approximating its second order derivative, we advise using Method I-3, Method II, and CSM [11].
When , Method I-1 and Method II are methods, Method I-2 is an method, and Method I-3 is an method. The cubic spline method (CSM) [11] is also an method, while the method in [12] is an method; the method in [13] is an method conditionally, only if the shape parameter therein. Obviously, the approximation orders of Method I-2 and Method I-3 are higher than the methods in [12, 13], the approximation orders of Method I-1 and Method II equal that of the method in [12], and the approximation order of Method I-3 equals that of CSM [11]. Undoubtedly, our methods are full of approximation ability. Furthermore, [12, 13] have not studied first order and second order derivative approximations. At the same time, our methods are more suitable for noisy data than the methods in [12, 13]. Hence, Method I-2, Method I-3, and CSM [11] are more preferable than others.
6. Conclusions
The explicit error bounds for a noisy lacunary cubic spline interpolation and a simple noisy cubic spline quasi interpolation are well studied in this paper; see Theorems 5 and 7. These new results are very useful in numerical approximation and related practical fields. Moreover, these results are also verified by some numerical examples. In a word, both theoretical analysis and numerical tests show that our methods are well behaved. We end the paper with the following remarks.(i)The main contributions of the paper include (i) studying two new methods to approximate a function and its first order and second order derivatives from the given noisy data and (ii) analyzing the explicit error bounds for the methods.(ii)The main advantages of our new methods include the following: (i) they are very simple; (ii) they are not only applicable to noisy data but also applicable to exact data; (iii) Method I-2 and Method I-3 have better performance in function approximation and first order derivative approximation than other methods; Method I-3 and Method II have better performance in second order derivative approximation than other methods.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors appreciate the reviewers and editors for their careful reading, valuable suggestions, and timely review and reply.