Multilevel anti-derivative wavelets with augmentation for nonlinear boundary value problems
- Somlak Utudee^{1} and
- Montri Maleewong^{2}Email author
https://doi.org/10.1186/s13662-017-1156-8
© The Author(s) 2017
Received: 12 August 2016
Accepted: 21 March 2017
Published: 4 April 2017
Abstract
The multilevel augmentation method with the anti-derivatives of the Daubechies wavelets is presented for solving nonlinear two-point boundary value problems. The anti-derivatives of the Daubechies wavelets are applied as the multilevel bases for the subspaces of approximate solutions. This process results in a full nonlinear system that can be solved by the multilevel augmentation method for reducing computational cost. The convergence rate of the present method is shown. It is the order of \(2^{s}\), \(0\leq s\leq p\) when p is the order of the Daubechies wavelets. Various examples of the Dirichlet boundary conditions are shown to confirm the theoretical results.
1 Introduction
Many problems in science and engineering can be modeled by nonlinear differential equations. Due to their complexities of both differential equation forms and boundary condition types, analytical solutions are available for only simple problems. Efficient and accurate numerical solutions are then usually required in general. One of the most effective numerical methods relies on variational formulations; see [1, 2], and [3].
The multilevel basis method can be applied with the variational formulation to obtain the approximate solutions of nonlinear problem. This formulation results in the discretization of nonlinear systems with unknown coefficients in the approximate subspace of each basis level. A nonlinear solver such as the Newton iterative method can be used to find approximate solutions for each level required. In order to obtain more accurate results, the number of applied basis levels must be increased, resulting in large nonlinear systems. The computational time increases exponentially with only a small increase in the basis levels. In order to reduce the computational time, we apply the advantage of multilevel bases by connecting the information among basis levels. This approach, the augmentation method, was first introduced by [1, 2, 4]. The fully nonlinear system is divided into two smaller systems and then solved separately.
In multi-scale decompositions, multi-scale piecewise polynomials can be applied in variational formulation (see [1, 4]). These basis types are easily presented and implemented as a numerical algorithm. They can be used for specific types of boundary conditions, and in this case they can represent Dirichlet conditions with zero boundary values while the modified approximation technique can be applied for non-zero Dirichlet conditions. To extend to a more general class of multilevel basis for solving various types of boundary conditions, the anti-derivatives of Daubechies wavelets introduced by Xu and Shann [5] can be applied to solve many kinds of boundary conditions: Dirichlet and Neumann types. Reference [6] has presented the case of the linear boundary value problem and shown that the Daubechies wavelets can be applied in conjunction with the augmentation method to save computational time for Dirichlet boundary value problem.
This study extends the multi-scale decomposition to a nonlinear boundary value problem. We apply the anti-derivatives of the Daubechies wavelets to solving nonlinear boundary value problems. The discretization of the nonlinear differential problem is represented by a nonlinear system that can be solved iteratively by the Newton method. To save computational time, the augmentation method presented by Chen, Chen, Wu and Xu (see e.g. [2, 4, 7, 8]) will be applied to solving the nonlinear system under the Daubechies wavelets. Combining these two concepts, as presented here, results in a new numerical method. The rate of convergence is also proved. It is of the order \(2^{s}\), \(0\leq s\leq p\) when p is the order of the Daubechies wavelets applied.
- 1.
Formulate the variational form of the considering problem. We determine approximate solution, if the variational form has a unique solution. Because it is not our main consideration in this work, we assume that the nonlinear boundary value problem and its variational form have the same isolated solution \(u^{*} \in H_{0}^{1}(a,b)\).
- 2.
Choose a sequence \(\{S_{n}\}\) of nested finite-dimensional subspaces of the solution space \(H_{0}^{1}(a,b)\) such that \(\overline {\bigcup_{n\in\mathbb{N}} S_{n}} = H_{0}^{1}(a,b)\). At this step, such finite-dimensional subspaces have anti-derivatives of Daubechies wavelets as their orthonormal bases.
- 3.
Apply the multilevel augmentation method (MAM) to obtain the nth level approximation, which is composed of two smaller systems. One is a linear system. Another one is a nonlinear system. The nonlinear system will be solved iteratively using the Newton method.
This work is organized as follows. In Section 2, we introduce the anti-derivatives of the Daubechies wavelets and the finite-dimensional subspaces of the solution subspaces of \(H^{1}(a,b)\). The concept of multilevel augmentation method is presented in Section 3. The estimations of the optimal error rate are shown in Section 4. Some numerical examples are demonstrated in Section 5. Conclusions are finally drawn in Section 6.
2 Bases for subspaces of \(H^{1}_{0}(a,b)\)
To apply our method for solving the nonlinear boundary value problem, we construct a sequence \(\{S_{n}\}\) of nested finite-dimensional subspaces of the solution space such that \(\bigcup_{n\in\mathbb{N}} S_{n}\) is dense in the solution space. In this section, we will give a brief introduction to the anti-derivatives of wavelets that are the orthonormal bases for the finite-dimensional subspaces of the solution space \(H^{1}_{0}(a,b) \).
The wavelets \(\{\psi_{jk} | j\geq-1, k\in I_{j} \}\) form a frame for \(L^{2}(a,b)\), that is, the set consisting of all linear expansions is equal to \(L^{2}(a,b)\).
In [5], Xu and Shann introduced the anti-derivatives of the Daubechies wavelets that form orthonormal bases for the finite-dimensional subspaces of solution spaces.
3 Multilevel augmentation method
In this section, we summarize the main concepts of multilevel augmentation method for solving the nonlinear boundary value problems. Readers can refer to [2, 4, 7, 8] for details.
It should be noted that the original full nonlinear system of \(\dim S_{n+i+1} \) can be solved in the augmentation method by just solving the smaller nonlinear system of the fixed size \(\dim S_{n}\). The increasing number of unknown coefficients when increasing the level approximation can be solved by the corresponding linear systems. Specially for our presented orthonormal basis, the linear system is easy to solve. The unknown coefficients in the higher level are obtained directly. Overall, the computational time can then be reduced greatly by this method.
Algorithm: The multilevel augmentation method based on the Galerkin method.
Let \(n, i\) be two fixed positive integers.
Step 4: Set \(m \leftarrow m+1\) and go back to Step 2 until \(m=i\).
The computational complexity which is measured by the number of multiplications and functional evaluations used in the computation of the above algorithm is of the order \(\mathcal{O}(\dim S_{n+m})\). More details of complexity analysis can be found in [4] and [2].
4 Error analysis
In this section, we will show the convergent rate of the multilevel augmentation method in conjunction with the anti-derivatives of the Daubechies wavelets. Let \(u^{*}\) be the isolated solution of (2), \(u_{n}\in S_{n}\) be the nth (standard) multilevel solution obtained from the wavelets of order p and \(u_{n, i}\) be the \((n+i)\)th multilevel augmentation solution of (2).
- (i)\(\varphi(x,u)\) is a real continuous function in \((x,u) \in[a,b] \times\mathbb{R}\), and satisfies the Lipschitz condition with respect to u for \(|u|\leq R, R\geq0\), that is,for some positive constant \(M_{1}\).$$\bigl\vert \varphi(x,v)-\varphi(x,v) \bigr\vert \leq M_{1}|u-w|,\quad |v|\leq R, |w|\leq R, $$
- (ii)\(\varphi(x,u)\) is continuously differentiable with respect to u for all \(x\in[a,b]\), and all \(v \in B(u^{*},\rho):=\{v | |v-u^{*}|\leq\rho\}\), for some \(\rho>0\), and there exists a positive constant \(M_{2}\) such that$$\bigl\vert \varphi_{u}(x,v)-\varphi_{u}(x,w) \bigr\vert \leq M_{2}|v-w|, \quad\text{for all } v, w \in B\bigl(x^{*},\rho\bigr). $$
The following lemma was proved in Section 3 of [4] but for convenience of the reader, we have reproduced and included its proof for ready reference.
Lemma 4.1
Proof
Next, we consider the difference between the isolated solution \(u^{*}\) and the \((n+i)\)th multilevel augmentation solution, \(u_{n, i}\), of (1).
Theorem 4.2
Proof
The variational form of (1) can be written in the form of \((\mathcal{I}-\mathcal{K})u =0\). By Lemma 4.1, the operator \(\mathcal{K}\) is completely continuous and Fréchet differentiable on the closed ball \(B(u^{*},\rho)\) and the Fréchet derivative \(\mathcal{K}'\) satisfies the Lipschitz condition.
The above estimation suggests that, if we apply the wavelet of order p, the solution \(u\in H^{1}_{0}(a,b) \cap H^{s+1}(a,b) \). If we apply the multilevel augmentation method from level \(n+i-1\) to \(n+i\) by the anti-derivatives wavelets of order p, the errors measured in \(L^{2}\)-norm decrease at most by a factor of \(2^{p}\). Consequently, the errors obtained by the standard multilevel and the multilevel augmentation methods decrease with the same order.
5 Numerical examples
In this section, we illustrate the accuracy of the multilevel augmentation method in conjunction with the anti-derivatives of the Daubechies wavelets of order p for solving nonlinear boundary value problems with Dirichlet boundary conditions.
Example 5.1
Next, we will show the calculation steps of the multilevel augmentation method. Assume that we have obtained \(u_{2}\in S_{2}\): \(u_{2}= b_{00}\overline{\Psi}_{00} + b_{10}\overline{\Psi}_{10} + b_{11}\overline{\Psi}_{11}\) from the second level of the standard multilevel method. The third level of approximation can be obtained by the multilevel augmentation method as follows.
Since we have to calculate the inner product of functions and bases, we perform it numerically by the trapezoidal rule in all of the examples. The derivatives are approximated using the central difference formula.
Numerical results for \(\pmb{p=1}\)
n | \(\dim S_{n}\) | \(\Vert u-u_{n} \Vert \) | \(\text{Time}_{n}\) | \(\Vert u-u_{1,n-1} \Vert \) | \(\text{Time}_{1,n-1}\) | \(\Vert u-u_{2,n-2} \Vert \) | \(\text{Time}_{2,n-2}\) | \(\Vert u-u_{3,n-3} \Vert \) | \(\text{Time}_{3,n-3}\) |
---|---|---|---|---|---|---|---|---|---|
1 | 3 | 2.3549e–1 | 3.5800e–2 | 2.3962e+0 | 8.3000e–3 | ||||
2 | 7 | 1.0022e–1 | 2.3140e–1 | 1.3037e+0 | 3.7700e–1 | 1.0073e+0 | 1.7500e–2 | ||
3 | 15 | 4.8223e–2 | 1.7408e+0 | 4.5435e–1 | 6.2230e–1 | 4.3987e–1 | 4.4840e–1 | 4.3458e–1 | 5.6600e–2 |
4 | 31 | 2.4206e–2 | 1.6873e+1 | 2.1642e–1 | 7.3150e–1 | 2.1044e–1 | 7.3400e–1 | 2.0832e–1 | 4.9160e–1 |
5 | 63 | 1.3392e–2 | 1.3638e+2 | 1.1440e–1 | 8.4620e–1 | 1.1187e–1 | 8.3810e–1 | 1.1097e–1 | 8.2520e–1 |
Numerical results for \(\pmb{p=2}\)
n | \(\dim S_{n}\) | \(\Vert u-u_{n} \Vert \) | \(\text{Time}_{n}\) | \(\Vert u-u_{1,n-1} \Vert \) | \(\text{Time}_{1,n-1}\) | \(\Vert u-u_{2,n-2} \Vert \) | \(\text{Time}_{2,n-2}\) | \(\Vert u-u_{3,n-3} \Vert \) | \(\text{Time}_{3,n-3}\) |
---|---|---|---|---|---|---|---|---|---|
1 | 7 | 9.6680e–5 | 3.8360e–1 | 7.9852e–2 | 2.4100e–2 | ||||
2 | 13 | 1.2588e–5 | 1.9599e+0 | 5.1073e–3 | 4.5320e-1 | 5.1347e–3 | 4.5900e–2 | ||
3 | 25 | 2.2134e–06 | 1.2856e+01 | 1.6628e–04 | 6.4740e–1 | 1.8461e–4 | 5.1630e–1 | 1.8573e–4 | 1.2630e–1 |
The \(L^{2}\) errors by the augmentation method are of the same order as those of the standard multilevel method, when applied at the same level. When comparing the results between \(p=1\) and \(p=2\), the \(L^{2}\) errors for \(p=2\) decrease faster than those for \(p=1\). The rate of convergence decreases by the factor of C2^{2}, for some constants C. The results from Tables 1 and 2 confirm the theoretical results of our main theorem.
Example 5.2
Numerical results for \(\pmb{p=1}\)
n | \(\dim S_{n}\) | \(\Vert u-u_{n} \Vert \) | \(\text{Time}_{n}\) | \(\Vert u-u_{1,n-1} \Vert \) | \(\text{Time}_{1,n-1}\) | \(\Vert u-u_{2,n-2} \Vert \) | \(\text{Time}_{2,n-2}\) | \(\Vert u-u_{3,n-3} \Vert \) | \(\text{Time}_{3,n-3}\) |
---|---|---|---|---|---|---|---|---|---|
1 | 3 | 2.3184e+0 | 3.5800e–2 | 2.3962e+0 | 8.3000e–3 | ||||
2 | 7 | 9.8550e–1 | 2.3140e–1 | 1.3037e+0 | 3.7700e–1 | 1.0073e+0 | 1.7500e–2 | ||
3 | 15 | 4.3358e–1 | 1.7408e+0 | 4.5435e–1 | 6.2230e–1 | 4.3987e–1 | 4.4840e–1 | 4.3458e–1 | 5.6600e–2 |
4 | 31 | 2.0971e–1 | 1.6873e+1 | 2.1642e–1 | 7.3150e–1 | 2.1044e–1 | 7.3400e–1 | 2.0832e–1 | 4.9160e–1 |
5 | 63 | 1.1300e–1 | 1.3638e+2 | 1.1440e–1 | 8.4620e–1 | 1.1187e–1 | 8.3810e–1 | 1.1097e–1 | 8.2520e–1 |
In our last example, we will test our present method to solve the nonlinear boundary value problem with non-zero Dirichlet boundary conditions.
Example 5.3
Numerical results for \(\pmb{p=1}\)
n | \(\dim S_{n}\) | \(\Vert u-u_{n} \Vert \) | \(\text{Time}_{n}\) | \(\Vert u-u_{1,n-1} \Vert \) | \(\text{Time}_{1,n-1}\) | \(\Vert u-u_{2,n-2} \Vert \) | \(\text{Time}_{2,n-2}\) | \(\Vert u-u_{3,n-3} \Vert \) | \(\text{Time}_{3,n-3}\) |
---|---|---|---|---|---|---|---|---|---|
1 | 3 | 3.6441e+0 | 5.5200e–2 | 4.0184e+0 | 1.5600e–2 | ||||
2 | 7 | 1.6750e+0 | 3.4570e–1 | 2.0257e+0 | 4.5810e–1 | 1.7817e+0 | 3.6700e–2 | ||
3 | 15 | 7.4431e–1 | 2.7760e+0 | 9.7397e–1 | 7.0520e–1 | 8.1878e–1 | 4.3340e–1 | 7.6238e–1 | 1.5370e–1 |
4 | 31 | 3.6599e–1 | 2.1067e+1 | 4.8121e–1 | 7.5820e–1 | 3.9986e–1 | 6.5820e–1 | 3.7317e–1 | 6.2760e–1 |
5 | 63 | 1.9502e–1 | 1.6468e+2 | 2.4916e–1 | 9.5190e–1 | 2.0838e–1 | 8.5460e–1 | 1.9563e–1 | 1.1097e+0 |
6 Conclusions
This study extends the multi-scale decomposition to a nonlinear boundary value problem. We apply the anti-derivatives of the Daubechies wavelets of order p to solve nonlinear two-point boundary value problems. The augmentation method is employed in a variational formulation for multilevel constructions. The present method can reduce computational time when solving the discretization of the full nonlinear system. The nonlinear system from the standard multilevel method can be separated or augmented into two smaller systems. One is linear and the other is a nonlinear one that can be solved iteratively by the Newton method. The numerical accuracy can be improved by increasing the resolutions or the level of approximations. The rate of convergence was shown to be at most on the order of \(2^{p}\) where p is the order of the wavelet basis. We illustrate numerically in our examples that the \(L^{2}\) error decreases when the number of basis levels increases. The rate of convergence from our estimations has been confirmed by many examples. Due to its advantages, the anti-derivatives of the Daubechies wavelets can be used to solve various kinds of boundary conditions. We are extending this study to apply this basis type with the augmentation method for solving Neumann type and mixed boundary conditions, without any modifications in the assumed form of approximate solution; these results will be reported elsewhere.
Declarations
Acknowledgements
This research was supported by the Faculty of Science, Chiang Mai University as regards the first author, and the Faculty of Science, Kasetsart University as regards the second author.
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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