Wavelet multilevel augmentation method for linear boundary value problems
- Somlak Utudee^{1} and
- Montri Maleewong^{2}Email author
https://doi.org/10.1186/s13662-015-0464-0
© Utudee and Maleewong; licensee Springer. 2015
Received: 25 September 2014
Accepted: 8 April 2015
Published: 23 April 2015
Abstract
This work presents a new approach to numerically solve the general linear two-point boundary value problems with Dirichlet boundary conditions. Multilevel bases from the anti-derivatives of the Daubechies wavelets are constructed in conjunction with the augmentation method. The accuracy of numerical solutions can be improved by increasing the number of basis levels, but the computational cost also increases drastically. The multilevel augmentation method can be applied to reduce the computational time by splitting the coefficient matrix into smaller submatrices. Then the unknown coefficients in the higher level can be solved separately. The convergent rate of this method is \(2^{s}\), where \(1 \leq s \leq p+1\), when the anti-derivatives of the Daubechies wavelets order p are applied. Some numerical examples are also presented to confirm our theoretical results.
Keywords
MSC
1 Introduction
On the other hand, wavelets can be applied to discretize differential equations (see, e.g., [5, 6]). Related numerical methods with the applications of Haar and Legendre wavelets for solving boundary value problem are proposed by Siraj-ul-Islam et al. [7, 8]. The advantage of wavelet basis is its capability to approximate solutions of differential equations. The wavelet Galerkin method is one of the most powerful methods that can be used to solve ordinary and partial differential equations (see, e.g., [9–11], and [12]). In addition, the accuracy of the approximate solutions can easily be improved by merely increasing the numbers of wavelet basis functions and the orders of wavelets. However, the wavelet basis is not straightforwardly adjusted to satisfy general boundary conditions. In 1992, Xu and Shann introduced a different approach to handle the boundary conditions by using the anti-derivatives of Daubechies wavelets [13]. These anti-derivatives form bases for the finite-dimensional subspaces of Sobolev space \(H^{1}\) and are used to construct an algorithm for approximating solutions.
In this work, we propose the method that combines the main advantages of wavelet bases and multilevel augmentation together. That is, we apply the multilevel augmentation of operators in conjunction with the anti-derivatives of Daubechies wavelets to approximate linear differential equations in the case of Dirichlet boundary conditions. The originality of this work is that we introduce the anti-derivatives of Daubechies wavelets for solving linear boundary value problems (see [13]) and apply this basis type with the augmentation method proposed by Chen (see, e.g., [1–3] and [4]). By this concept, we obtain a new approach to reduce the computational time for solving the linear system resulting from discretizing a linear differential equation.
Let \(p\in\mathbb{N}\). We will apply the multilevel augmentation method and anti-derivatives of wavelets of order p to find numerical solutions of two-point boundary value problems with Dirichlet boundary conditions.
- 1.
The solution space \(H^{1}_{0}(\varOmega )\) is decomposed into orthogonal direct sum of subspaces. The anti-derivatives of the Daubechies wavelets are used to construct finite-dimensional subspaces.
- 2.
For \(n\in\mathbb{N}\), the multilevel method is applied to obtain the nth level solution by solving a linear system with matrix coefficients related to the anti-derivatives of the Daubechies wavelets.
- 3.
To obtain a solution at a higher level, namely \((n+i)\) th level, the multilevel augmentation method is applied. By the algorithm to be presented, the computational time for solving the linear system is reduced since the dimension of the matrix coefficient is smaller.
Finally, this work is organized as follows. Section 2 gives an introduction to the anti-derivatives of the Daubechies wavelets and the finite-dimensional subspaces of the solution space \(H^{1}_{0}(\varOmega )\). In Section 3, we describe the algorithm to find approximate solutions using the multilevel augmentation method. The optimal error estimates for the approximate solutions are proven in Section 4, while some numerical examples are demonstrated in Section 5. Conclusions and future work are discussed in Section 6.
2 Bases for subspaces of \(H_{0}^{1}(\varOmega )\)
In this section, we will introduce the wavelets of order p and their anti-derivatives. These functions form orthonormal bases for the finite-dimensional subspaces \(S_{n}\) of the solution space \(H^{1}_{0}(\varOmega )\). More details can be found in [5] and [13].
To define the Daubechies wavelets, we consider two functions: the scaling function \(\phi(x)\) and the wavelet function \(\psi (x)\). The scaling function is obtained from the dilation equation. The wavelet function is defined from the scaling function. Details are described as follows.
\(\{\psi_{jk}|_{\varOmega }\mid j \geq-1, k\in I_{j} \}\) is a frame of \(L^{2}(\varOmega )\). That is, the \(\operatorname {span}\{\psi_{jk}|_{\varOmega }\mid j \geq-1, k\in I_{j} \}\) consisting of all linear expansions is equal to \(L^{2}(\varOmega )\).
3 Multilevel augmentation method algorithm
In this section, we describe the multilevel augmentation method for solving boundary value problems.
4 Error analysis
Next, we consider the distance between the solution u and the \((n+i)\)th multilevel augmentation solution, \(u_{n, i}\), of (1). In the remaining section, we denote by \(\mathcal{A}\) the operator corresponding to the matrix A, and we denote by u the column matrix representing element u.
Theorem 1
(Error for multilevel augmentation method)
Proof
The above theorem suggests that, if the solution \(u\in H_{0}^{1}(\varOmega ) \cap H^{s}(\varOmega ) \), and we apply the multilevel augmentation method from level \(n+i-1\) to \(n+i\) by using the anti-derivatives wavelets of order p, the errors measured in \(\|\cdot\|\) decrease by a factor of \(2^{p+1}\). Thus the behaviors of the decreasing error obtained by the multilevel, and the multilevel augmentation methods, are in the same order.
5 Examples
We assume that \(f\in L^{2}(\varOmega )\), the coefficients q and r are smooth in the closed interval \([0,R]\) with \(q > 0\) and \(r \geq0\).
Example 1
Example 1 : Numerical results for \(\pmb{p=1}\)
n | \(\boldsymbol {\dim S_{n}}\) | \(\boldsymbol {\|u-u_{n}\|}\) | \(\boldsymbol {\|u-u_{1,n-1}\|}\) | \(\boldsymbol {\|u-u_{2,n-2}\|}\) |
---|---|---|---|---|
2 | 3 | 8.0368e−001 | ||
3 | 7 | 5.6412e−001 | 5.7409e−001 | |
4 | 15 | 3.7341e−001 | 3.9796e−001 | 4.0020e−001 |
5 | 31 | 2.2508e−001 | 2.7097e−001 | 2.6985e−001 |
6 | 63 | 1.4568e−001 | 1.7275e−001 | 1.7293e−001 |
7 | 127 | 8.8603e−002 | 1.1148e−001 | 1.1143e−001 |
Example 2
Example 2 : Numerical results for \(\pmb{p=1}\)
n | \(\boldsymbol {\dim S_{n}}\) | \(\boldsymbol {\|u-u_{n}\|}\) | \(\boldsymbol {\|u-u_{1,n-1}\|}\) | \(\boldsymbol {\|u-u_{2,n-2}\|}\) | \(\boldsymbol {\|u-u_{3,n-3}\|}\) |
---|---|---|---|---|---|
2 | 3 | 1.6427 | |||
3 | 7 | 0.6045 | 0.6458 | ||
4 | 15 | 0.2424 | 0.2488 | 0.2494 | |
5 | 31 | 0.1124 | 0.1162 | 0.1162 | 0.1170 |
6 | 63 | 0.0589 | 0.0607 | 0.0607 | 0.0606 |
7 | 127 | 0.0309 | 0.0325 | 0.0325 | 0.0325 |
8 | 255 | 0.0154 | 0.0161 | 0.0161 | 0.0161 |
9 | 511 | 0.0085 | 0.0087 | 0.0087 | 0.0087 |
Example 2 : Numerical results for \(\pmb{p=2}\)
n | \(\boldsymbol {\dim S_{n}}\) | \(\boldsymbol {\|u-u_{n}\|}\) | \(\boldsymbol {\|u-u_{1,n-1}\|}\) |
---|---|---|---|
0 | 4 | 1.6268 | |
1 | 7 | 0.2097 | |
2 | 13 | 0.0250 | 0.0341 |
3 | 25 | 0.0030 | 0.0041 |
4 | 49 | 0.0004 | 0.0005 |
Example 3
Example 3 : Numerical results for \(\pmb{p=1}\)
n | \(\boldsymbol {\dim S_{n}}\) | \(\boldsymbol {\|u-u_{n}\|}\) | \(\boldsymbol {\|u-u_{1,n-1}\|}\) | \(\boldsymbol {\|u-u_{2,n-2}\|}\) | \(\boldsymbol {\|u-u_{3,n-3}\|}\) |
---|---|---|---|---|---|
2 | 3 | 0.4221 | |||
3 | 7 | 0.1993 | 0.2123 | ||
4 | 15 | 0.0994 | 0.1051 | 0.1049 | |
5 | 31 | 0.0502 | 0.0561 | 0.0555 | 0.0555 |
6 | 63 | 0.0278 | 0.0303 | 0.0303 | 0.0302 |
Example 3 : Numerical results for \(\pmb{p=2}\)
n | \(\boldsymbol {\dim S_{n}}\) | \(\boldsymbol {\|u-u_{n}\|}\) | \(\boldsymbol {\|u-u_{1,n-1}\|}\) |
---|---|---|---|
0 | 4 | 3.3770 | |
1 | 7 | 0.2888 | |
2 | 13 | 0.0351 | 0.0378 |
3 | 25 | 0.0044 | 0.0047 |
4 | 49 | 0.0007 | 0.0008 |
6 Conclusions
This present work is our attempt to apply the multilevel augmentation method using the anti-derivatives of Daubechies wavelets for approximating two-point boundary value problems with Dirichlet boundary conditions. This method is extended from the multilevel augmentation method that uses polynomial wavelet basis. An error analysis has also been presented. The rate of convergence is by a factor of \(2^{s}\), \(1 \le s \le p+1\), where p is the Daubechies wavelet order. At the same level, the \(L^{2}\) error of the multilevel augmentation method is greater than that of the multilevel method, but they are in the same order.
The difficulty of this approach is that the anti-derivatives of Daubechies wavelet cannot be expressed in explicit form. One is required to solve the dilation equation to obtain a wavelet basis in an implicit formula. Here, we have done this using a numerical approximation to obtain the basis function point by point. Also, it is not easy to extend this approach to problems in higher dimension.
The advantage of this method is that we need not solve the full linear system. The unknown coefficients from the previous level can be used to approximate additional unknowns in the next level. Thus, this method can reduce computational time and memory for storing matrix coefficients. Furthermore, by applying the general anti-derivatives of Daubechies wavelets, this method can be used to solve the boundary value problems with Neumann and mixed boundary conditions. Numerical experiments on these problems are in progress and will be reported elsewhere further.
Declarations
Acknowledgements
This research was supported by Chiang Mai University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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