4. Some examples of application of wavelets to PDE

By now it should be clear how multiresolution analysis can be applied to differential equation solution: by using scaling functions and/or wavelets as basis functions in approximation (6). All the approaches described in Section 2 can be used with wavelet: collocation, Galerkin PDE method, weighted residual method, meshless methods, etc. Before describing some details of few approaches described in the literature, it is worth to do some general remarks.

What makes wavelet interesting is their multiresolution property and the fact that a wisely chosen wavelet (smooth and/or with many vanishing gradients) has interesting "singularity sensing" properties: in the neighborhood of a singularity (discontinuity, nondifferentiability, etc.), the coefficients decay as a function of scale with a speed that depends on the singularity involved (similar to what Fourier transform does, only on a local level), but away from the singularity, they decade fast [3]. This implies that good approximations can be obtained with few coefficients, using high-resolution decomposition only where it is necessary, reducing the size of the matrices involved in the solution of the PDE. A similar effect can be obtained, for example, in FEM by using a finer mesh around points of large variation. However, using this approach in an adaptive way would require to adjust at running time the mesh, a potentially heavy operation. Wavelets have the potential of employing an adaptive resolution in an easier way. See, for example, [10] for few examples of adaptive techniques employing wavelets.

While orthogonality is considered an important feature in many theoretically wavelet papers, in the context of differential equation solution, it plays a smaller role. The reason is that basis functions enter in the scalar products associated with the various methods via the differential operator D, and it is not guaranteed that D will preserve orthogonality (that would give rise to many zero entries, that is, sparser matrices).

Actually, orthogonality is preserved if the two basis functions have disjoint support in space (since differential operators do not extend the support) or in frequency (since differential operators are translation-invariant and in frequency they become a product). This suggests that in the context of differential equations, compact support and well-localization in frequency are more important than just orthogonality. In a sense, they represent a "robust" orthogonality condition.

### Remark 4.1.

It is true that true compact support in frequency is less common than compact support in space. With the exception of few very special and theoretical cases (e.g., Sinc), the best we can get is a rapid decay in frequency. This means that the scalar product of two basis functions separated in frequency will be maybe very small, but not zero. Nevertheless, even this kind of "almost sparseness" can be exploited.

A general issue with wavelets is that it can be difficult to impose boundary conditions since they have no natural interpolation property that would make

3.2.2 Second-generation wavelets

Wavelet Transform and Complexity

Figure 2.

Figure 3.

48

generation wavelet has been introduced.

wavelet (third row) of three different Daubechies' wavelets.

Two-scale Eq. (37) and the resulting filter bank-based procedure work well when the data are sampled on a regular grid and/or the functions of interest are defined on IR<sup>d</sup>. Since there are many applications that do not satisfy this requirement (e.g., differential equations on general manifolds), the idea of second-

Daubechies' wavelets. First three iterations of O (first row), the final scaling function (second row), and the

The starting point is the so-called lifting form of filter bank (Figure 1). It is possible to show that any two-channel filter bank (Figure 1) can be implemented as shown in Figure 3. In the lifting approach, the input signal is split into odd and even samples by a serial-to-parallel converter. The first branch is filtered, and the result combined with the other branch; the result of this operation is filtered again and combined with the first branch, and this iterated as long as necessary. Filter P is

Lifting implementation of the two-channel filter banks associated with a wavelet analysis.

boundary condition handling simpler. Another problem that can arise is that many wavelets have no closed-form description, for example, Daubechies wavelets that are described as the result of the iteration of operator O in (39). This can make their application to PDE more difficult, for example, when computing scalar products involved in weighted residual and other methods.

Finally, another common issue is that most of the known wavelets are defined on a one-dimensional domain, while many physical systems are on a multidimensional domain. The easiest way to create a multidimensional multiresolution analysis by a one-dimensional one is the separable (or tensor product or Kronecker product) approach that create a multidimensional function by the product of several one-dimensional ones, e.g.,

$$
\phi\_{3D}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \phi(\mathbf{x})\phi(\mathbf{y})\phi(\mathbf{z})\tag{61}
$$

to FEM techniques is the difficulty to adapt the wavelet construction to complex meshes. In this case the use of second-generation wavelet based on an extension of

[9, 48, 49]. Of some interest for the special problem is [9] that uses a wavelet approach to implement a meshless solver for differential equations defined on the sphere. A problem with applying wavelets to generic manifolds like a sphere is that it is not clear what a "rescaling by 2" should mean for a manifold that is not a Euclidean space. The idea used in [9] is to use a so-called diffusion wavelet where the dilation is replaced by a diffusion operator that looks like a kind of "low-pass filter-

This chapter introduced the reader to the field of applying wavelets to the numerical solution of differential equations. Both wavelets and differential equations are research fields with many applications, contributions, and results. Their combination gives rise to wide varieties of methods, each one suited for specific applications. By looking at the literature, we can see that wavelets can be a very powerful tool for solving PDE especially because of their multiresolution nature that allows to optimize the level of detail where it is needed. Wavelets, however, are not a silver bullet for all problems either, since they can have some characteristics (multidimensional construction via tensor product, nonexistence of a closed-form expression, difficulty in handling some boundary conditions, etc.) that can make their application not trivial in some cases. We can say that this is a field where, more than ever, no single solution fits all and that every practitioner needs to find the solution specific for the problem at hand using knowledge in both fields and some

ing" that smear out the details; see [9] for the precise definition.

Wavelets have attracted some interest also in the context of meshless methods

the lifting idea has attracted some attention [45–47].

DOI: http://dx.doi.org/10.5772/intechopen.82820

Wavelets for Differential Equations and Numerical Operator Calculus

5. Conclusions

ingenuity.

Author details

51

Riccardo Bernardini

University of Udine, Udine, Italy

provided the original work is properly cited.

\*Address all correspondence to: riccardo.bernardini@uniud.it

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

This kind of approach, however, produces "cube-like" wavelets, and their application to FEM schemes based on triangular elements can be difficult.

### 4.1 Some schemes from the literature

In this section we briefly summarize some interesting wavelet-based schemes that can be found in the literature. As said above, wavelets and scaling functions can be used as basis in the approximation used in collocation, weighted residuals, and other methods.

Wavelets in Galerkin and weighted residual methods bring the advantage of their multiresolution and localization properties while, however, suffering from difficulties in handling complex boundary conditions. Moreover, nonlinear equations can turn out to be difficult to handle. Nevertheless, there have been many successful examples in the application to elliptic, hyperbolic, and parabolic PDE [11–27]. Wavelet-based collocation methods, where wavelet functions are used as shape functions, also registered some success. The advantage of collocation methods is that they are more easily applicable in nonlinear cases [28, 29] and irregular boundary conditions [30]. A collocation method based on second-generation wavelet and lifting is applied to a nonlinear vibration problem in [30, 31].

Much more popular seems to be the application of wavelets to FEM techniques. In this case wavelets or scaling functions are used as shape functions instead of the more traditional polynomials. Daubechies wavelets are particularly popular most probably because of their compact support property. Also of interest is the fact that Daubechies' wavelets can have any number of null moments, making possible the perfect interpolation of polynomials. Some examples of successful application Daubechies wavelets to PDE (mostly mechanical problems) are [32–36]. Of special interest is the proposal of Mitra [37] where wavelet-based FEM is used to transform a wave propagation problem into ordinary differential equations that are successively solved.

Another popular solution for wavelet-based FEM is the wavelets based on spline spaces. Although spline bases cannot have both compact support and orthogonality, in differential equations, as explained above, we gladly give up on orthogonality if we can get compact support and smoothness. Another important advantage of splines is that a simple closed-form expression is known. Examples of spline applications can be found in [38–40]. Of special interest is the application of Hermite cubic splines (HCS), a kind of multiwavelet [41] that shows promise in handling in a numerically robust way boundary conditions. The HCS is a multiwavelet with four smooth (twice differentiable) components defined on interval 0½ � ; 1 . Some examples of application can be found in [42–44]. A problem with the application of wavelets

Wavelets for Differential Equations and Numerical Operator Calculus DOI: http://dx.doi.org/10.5772/intechopen.82820

to FEM techniques is the difficulty to adapt the wavelet construction to complex meshes. In this case the use of second-generation wavelet based on an extension of the lifting idea has attracted some attention [45–47].

Wavelets have attracted some interest also in the context of meshless methods [9, 48, 49]. Of some interest for the special problem is [9] that uses a wavelet approach to implement a meshless solver for differential equations defined on the sphere. A problem with applying wavelets to generic manifolds like a sphere is that it is not clear what a "rescaling by 2" should mean for a manifold that is not a Euclidean space. The idea used in [9] is to use a so-called diffusion wavelet where the dilation is replaced by a diffusion operator that looks like a kind of "low-pass filtering" that smear out the details; see [9] for the precise definition.
