Wavelet Based Lifting Schemes for the Numerical Solution of Parabolic Partial Differential Equations
S. C. Shiralashetti1*, L. M. Angadi2 and A. B. Deshi3
1P. G. Department of Studies in Mathematics, Karnatak University, Dharwad-580003, INDIA.
2Govt. First Grade College, Chikodi-591201, INDIA.
3Department of Mathematics, KLECET, Chikodi – 591201, INDIA.
*Corresponding author: email: [email protected] (Received on: February 21, 2018)
ABSTRACT
Many phenomena’s in engineering, biology, fluid mechanics and other sciences can be modeled as both linear and nonlinear partial differential equations (PDEs). In this paper, we presented wavelet based Lifting schemes for the numerical solution of parabolic partial differential equations using orthogonal and biorthogonal wavelet filter coefficients. The numerical results obtained by this scheme are compared with the exact solution to demonstrate the accuracy and also speeds up convergence in lesser computational time as compared with existing scheme. Some of the test problems are presented for the applicability and validity of the scheme.
Keywords: Orthogonal and Biorthogonal wavelets; Lifting scheme; Parabolic PDEs.
1. INTRODUCTION
Partial differential equations appear in a wide variety of scientific applications such as plasma physics, solid state physics, optical fibers, biology, fluid dynamics and chemical kinetics1. In last decade there has been much attention for finding the numerical solution of such type of partial differential equations (PDEs). In general PDEs with or without reaction terms are used as a fundamental tool to model a wide class of phenomena occurring in physical and biological sciences. Recently, some of the iterative methods are used for the numerical and analytical solutions of linear and nonlinear partial differential equations. For example, Adomian decomposition method2, He’s variational iteration technique3 and the homotopy perturbation method4, etc.
Analytical solution of certain parabolic PDEs either does not exist or is hard to find.
Due to this fact, in the last decades, there have been great advances in the development of finite difference, finite element, spectral techniques and finite volume methods for the solution of parabolic PDEs. The parabolic PDEs of the form5,
ut uxx f u( ) or ut uxxg x t( , ), 0 x 1,t0 (1.1) subject to initial condition (IC) and boundary conditions (BCs). Where f u( ) and g x t( , ) are
the functions of dependent and independent variables.
Wavelet analysis assumed significance due to successful applications in signal and image processing during the 1980s. The smooth orthonormal basis obtained by the translation and dilation of a single function in a hierarchical fashion proved very useful to develop compression algorithms for signals and images up to a chosen threshold of relevant amplitudes. Some of the major contributors to this theory are: multiresolution signal processing used in computer vision; sub band coding, developed for speech and image compression; and wavelet series expansion, developed in applied mathematics.
Some of the earlier works on wavelet based methods can be found in6. A collection of the discrete wavelet transforms (DWT) and the FAS were introduced recently in7-8. The wavelet based full approximation scheme (WFAS) has exposed to be a very efficient and favorable method for numerous problems related to computational science and engineering fields9. These methods can be either used as an iterative solver or as a preconditioning technique, offering in many cases a better performance than some of the most innovative and existing FAS algorithms. Due to the efficiency and potentiality of WFAS, researches further have been carried out for its enrichment. In order to realize this task, work build that is orthogonal/biorthogonal discrete wavelet transform using lifting scheme10. Wavelet based lifting technique is introduced by Sweldens11, which permits some improvements on the properties of existing wavelet transforms. Wavelet based numerical solution of elasto- hydrodynamic lubrication problems via lifting scheme was introduced by Shiralashetti et al.12 The technique has some numerical benefits as a reduced number of operations which are fundamental in the context of the iterative solvers. Evidently all attempts to simplify the wavelet solutions for PDE are welcome. In PDE, matrices arising from system are dense with non-smooth diagonal and smooth away from the diagonal. This smoothness of the matrix transforms into smallness using wavelet transform and it leads to design the effective wavelets based lifting scheme.
Lifting scheme is a new approach to construct the so-called second generation wavelets that are not necessarily translations and dilations of one function. The latter we refer to as a first generation wavelets or classical methods. The lifting scheme has some additional advantages in comparison with the classical wavelets. This transform works for signals of an arbitrary size with correct treatment of boundaries. Another feature of the lifting scheme is that all constructions are derived in the spatial domain. This is in contrast to the traditional approach, which relies heavily on the frequency domain. The two major advantageous are:
It leads to a more intuitively appealing treatment better suited to those in interested in applications than mathematical foundations.
It makes a computational time optimal and sometimes increasing the speed of calculations.
The lifting scheme starts with a set of well-known filters, thereafter lifting steps are used an attempt to improve (lift) the properties of a corresponding wavelet decomposition.
This procedure has some mathematical benefits as a reduced number of operations which are essential in the context of the iterative solvers. In addition to this, the present paper illustrates that the application of the lifting scheme for the numerical solution of parabolic PDEs.
The present paper is organized as follows: Preliminaries of wavelet filter coefficients and lifting scheme is given in section 2, In Section 3 provides the method of solution.
Numerical results of the test problems are given in section 4 and finally, in section 5 conclusion of the proposed work is given.
2. PRELIMINARIES OF WAVELET FILTER COEFFICIENTS AND LIFTING SCHEME The important feature of the lifting scheme is that every filter bank based on lifting automatically satisfies perfect reconstruction properties.
The lifting scheme is alternate approach to constructing wavelets and computing wavelet transform. Unlike the classical wavelet construction methods demonstrated in the Fourier transform, the lifting procedure for discrete wavelet transform and for wavelet construction is carried out entirely in the time domain. Instead of using the filter banks for decomposition and reconstruction of a signal, the lifting procedure uses three steps - namely, the split [S], predict [P] and update [U] to decompose a signal into approximation and wavelet coefficients as given in the Figure 1.
Fig. 1. Steps in lifting scheme
Split: Splitting the signal into two disjoint sets of samples.
Predict: If the signal contains some structure, then we can expect correlation between a sample and its nearest neighbors. i.e. d oddP(even)
Update: Given an even entry, we have predicted that the next odd entry has the same value, and stored the difference. We then update our even entry to reflect our knowledge of the signal.
i.e. sevenU( )d
The general lifting stages for decomposition and reconstruction of a signal are given in Fig. 2.
Fig. 2. Lifting wavelet algorithm.
The main advantage of the lifting procedure is that the coefficients can be computed in place to minimize the storage requirement and the reconstruction phase can be carried out by running the decomposition process backward. The procedure eliminates the need for the Fourier transform as in the classical method for signal processing.
Some of the advantages are as follows:
Fourier techniques are not required.
The transform can be carried out without a fixed form for the P and U operators.
The computation of the inverse transform is straightforward.
There are many variations on the operators P and U that generate various wavelets.
Now, we have discussed about different wavelet filters as follows:
a) Haar wavelet filter coefficients
We know that low pass filter coefficients
0 1
1 1
, ,
2 2
T
a a T
and high pass filter coefficients
0 1
1 1
, ,
2 2
T
b b T
play an important role in decomposition. Thus it is natural to wonder that it is possible to model the decomposition in terms of linear transformations i.e. matrices. Moreover, since digital signals and images are composed of discrete data, we need a discrete analog of the decomposition algorithm so that we can process signal and image data.
b) Daubechies wavelet filter coefficients
Daubechies introduced scaling functions having the shortest possible support. The scaling function
N has support
0, N1
, while the corresponding wavelet
N has support in the interval
1N/ 2, N/ 2
.Fig. 2. Lifting wavelet algorithm.
sj
sj - 1
dj - 1
dj - 1
sj - 1
sj
We have low pass filter coefficients
0 1 2 3
1 3 3 3 3 3 1 3
4 2 4 2 4 2 4 2
, , , , , ,
T
a a a a T
and high pass filter coefficients
0 1 2 3
1 3 3 3 3 3 1 3
4 2 4 2 4 2 4 2
, , , , , ,
T
b b b b T
.
c) Biorthogonal (CDF (2,2)) wavelets
Let’s consider the (5, 3) biorthogonal spline wavelet filter pair, the low pass filter pair
are 1 0 1 1 1 1
( , , ) , ,
2 2 2 2 2
a a a
and 2 1 0 1 2
1 1 3 1 1
( , , , , ) , , , ,
4 2 2 2 2 2 2 2 4 2
a a a a a
.
But, we have bk ( 1)ka1k and bk ( 1)ka1k, the high pass filter pair are
0 1 2
1 1 1
, ,
2 2 2 2 2
b b b & 1 1 0 1 1 3 2 1 3 1
, , , ,
4 2 2 2 2 2 2 2 4 2
b b b b b
Foundations of lifting scheme
Consider to numbers a, b as two neighbouring samples of a sequence and then these have some correlation which we would like to take advantage. The simple linear transform which replaces a and b by average s and difference d i.e.
&
2 2
a b a b
s d
The idea is that if a and b are highly correlated, the expected absolute value of their difference d will be small and can be represented with fever bits. In case that a = b, the difference is simply zero. We have not lost any information because we can always recover a and b from the gives s and d as:
&
2 2
d d
a s b s
The detailed algorithm using different wavelets is given in the next section.
3. METHOD OF SOLUTION Consider the parabolic PDEs
ut uxx f u( ) or ut uxxg x t( , ), 0 x 1,t 0 (3.1) After discretizing the equation (3.1) through the finite difference method (FDM), we get
system of algebraic equations. Through this system we can write the system as
Aub (3.2)
where A is N N coefficient matrix, b is NN matrix and u is NN matrix to be determined. Where N2J, N is the number of grid points and J is the level of resolution.
Solve Eq. (3.2) through the iterative method, we get approximate solution.
Approximate solution containing some error, therefore required solution equals to sum of approximate solution and error. There are many methods to minimize such error to get the accurate solution. Some of them are HWLS, DWLS BWLS etc. Now we are using the advanced technique based on different wavelets called as lifting scheme. Recently, lifting schemes are useful in the signal analysis and image processing in the area of science and engineering. But currently it extends to approximations in the numerical analysis9. Here, we are discussing the algorithm of the lifting schemes as follows.
3.1 Haar wavelet Lifting scheme (HWLS)
In10, Daubechies and Sweldens have shown that every wavelet filter can be decomposed into lifting steps. More details of the advantages as well as other important structural advantages of the lifting technique can be available in11. The representation of Haar wavelet via lifting form presented as;
Decomposition
Consider approximate solution S Pj like as signal and then apply the HWLS decomposition (finer to coarser) procedure as,
1 1 1
2 2 1 2 1
1 1
1
, 1 ,
2
2 and 1
2
j j j
d S S s S d
S s D d
(3.3)
In this stage finally, we get new approximation as, [ 1 ]
S S D (3.4) Reconstruction
Consider Eq. (3.2) and then apply the HWLS reconstruction (coarser to finer) procedure as,
1 1
1
1 1 1
2 1 2 2 1
2 , 1 ,
2
1 and
j 2 j j
d D s S
S s d S d S
(3.5)
which is the required solution of the given equation.
3.2 Daubechies wavelet Lifting scheme (DWLS)
As discussed in the previous section 3.1, we follow the same procedure but we used different wavelet i.e., Daubechies 4th order wavelet coefficient. The DWLS procedure is as follows;
Decomposition
1
2 1 2
1 1 1
2 1
2 1 1
1
2 1
1
3 ,
3 3 2
4 4 ,
, 3 1
and 2
3 1 2
j j
j j
j
s S S
d S s s
s s d
S s
D d
(3.6)
Here, we get new approximation as, [ 1 ]
S S D (3.7) Reconstruction
Consider Eq. (3.5), then apply the DWLS reconstruction (coarser to finer) procedure as,
1
2
1
2 1
1 1
1 1
2 1 1
1
2 1 2
2 ,
3 1
2 ,
3 1 ,
3 3 2
4 4 and
3
j j
j j
j
j j
d D
s S
s s d
S d s s
S s S
(3.7)
which is the required solution of the given equation.
3.3 Biorthogonal wavelet Lifting scheme (BWLS)
As discussed in the previous sections 3.1 and 3.2, we follow the same procedure here we used another wavelet i.e., biorthogonal wavelet (CDF(2,2)). The BWLS procedure is as follows;
Decomposition
1
2 2 1 2 2
1 1 1
2 1 1
1
1 1
1 ,
2
1 ,
4
1 ,
2 2
j j j
j j
d S S S
s S d d
D d
S s
(3.8)
In this stage finally, we get new signal as,
[ 1 ]
S S D (3.9) Reconstruction
Consider Eqn. (3.9), then apply the DWLS reconstruction (coarser to finer) procedure as
1
1 1
1 1 1
2 1 1
1
2 2 1 2 2
1 ,
2 2 ,
1 4
1 ) ,
2
j j
j j j
s S
d D
S s d d
S d S S
(3.10)
which is the required solution of the given equation.
The coefficients s1 j and d1 j are the average and detailed coefficients respectively of the approximate solution u . The new approaches are tested through some of the numerical a problems and the results are shown in next section.
4. NUMERICAL ILLUSTRATION
In this section, we applied Lifting scheme for the numerical solution of Fisher’s equation and also show the capability and applicability of HWLS, DWLS and BWLS. The error is computed by Ema x maxueua , where u and e u are exact and approximate a solution respectively.
Test Problem 1: Now consider the homogeneous linear diffusion equation
, 0 1, 0
ut uxxu x t
subject to the initial conditions
u x ( , 0) x e
xand boundary conditions 1
(0, ) 1 & (1, ) t
u t u t e e , the exact solution of the problem is u x t( , )xet ex 5.
By applying the methods explained in the section 3, we obtain the numerical solutions and compared with exact solution are presented in figure 1. The maximum absolute errors with CPU time of the methods are presented in table 1.
NN = 64 NN= 256
Fig. 1. Comparison of numerical solutions with exact solution of test problem 4.1 for NN=64 & 256.
Table 1. Maximsum error and CPU time (in seconds) of the methods of test problem 4.1.
NN Method Emax Setup time Running time Total time
44
FDM 4.1408e-03 3.0652 0.0756 3.1408
HWLS 4.1408e-03 0.0011 0.0021 0.0032
DWLS 4.1408e-03 0.0009 0.0105 0.0114
BWLS 4.1408e-03 0.0008 0.0043 0.0051
8 8
FDM 2.3758e-03 2.6823 0.0892 2.7715
HWLS 2.3758e-03 0.0011 0.0021 0.0032
DWLS 2.3758e-03 0.0009 0.0100 0.0109
BWLS 2.3758e-03 0.0008 0.0043 0.0051
16 16
FDM 1.2959e-03 2.8389 0.1514 2.9903
HWLS 1.2959e-03 0.0011 0.0021 0.0032
DWLS 1.2959e-03 0.0010 0.0100 0.0110
BWLS 1.2959e-03 0.0008 0.0044 0.0052
32 32 FDM 6.8050e-04 4.9817 1.1174 6.0991
HWLS 6.8050e-04 0.0011 0.0021 0.0032
DWLS 6.8050e-04 0.0009 0.0101 0.0110
BWLS 6.8050e-04 0.0008 0.0044 0.0052
Test problem 4.2. Next, consider the non-homogeneous linear diffusion equation
cos , 0 1, 0
ut uxx x x t subject to the initial conditions u x( , 0) 0 and boundary conditions (0, ) 1 t & (1, ) cos(1) (1 t)
u t e u t e , the exact solution
of the problem is u x t( , ) cos x (1 et) 5. By applying the methods explained in the section 3, we obtain the numerical solutions and compared with exact solution are presented in figure 2. The maximum absolute errors with CPU time of the methods are presented in table 2.
NN = 64 NN= 256
Fig. 2. Comparison of numerical solutions with exact solution of test problem 4.2 for NN=64 & 256.
Table 2. Maximum error and CPU time (in seconds) of the methods of test problem 4.2 NN Method Emax Setup time Running time Total time
44
FDM 6.9470e-03 2.9842 0.0754 3.0596
HWLS 6.9470e-03 0.0011 0.0021 0.0032
DWLS 4.1792e-03 0.0009 0.0102 0.0111
BWLS 6.9470e-03 0.0008 0.0022 0.0030
8 8
FDM 4.1792e-03 3.4513 0.0878 3.5391
HWLS 4.1792e-03 0.0011 0.0020 0.0031
DWLS 4.1792e-03 0.0010 0.0102 0.0112
BWLS 4.1792e-03 0.0008 0.0045 0.0053
16 16
FDM 2.3120e-03 3.3247 0.1410 3.4657
HWLS 2.3120e-03 0.0011 0.0020 0.0031
DWLS 2.3120e-03 0.0009 0.0100 0.0109
BWLS 2.3120e-03 0.0008 0.0044 0.0052
32 32 FDM 1.2212e-03 4.9140 0.9496 5.8636
HWLS 1.2212e-03 0.0011 0.0021 0.0032
DWLS 1.2212e-03 0.0009 0.0101 0.0110
BWLS 1.2212e-03 0.0008 0.0044 0.0052
Test problem 5.3. Finally, Consider the nonlinear Parabolic PDE
t x xx
u u u u
subject to the I.C.: u x
, 0 x and B.C.s: u
0,t 0, u
1,t 1
1t
which has the exact solution u x t( , ) x
1t
13. By applying the methods explained in thesection 3, we obtain the numerical solutions and compared with exact solution are presented in figure 3. The maximum absolute errors with CPU time of the methods are presented in table 3.
NN = 64 NN= 256
Fig. 3. Comparison of numerical solutions with exact solution of test problem 4.3 for NN=64 & 256.
Table 3. Maximum error and CPU time (in seconds) of the methods of test problem 4.3 NN Method Emax Setup time Running time Total time
44
FDM 5.4621e-03 3.2655 0.0008 3.2663
HWLS 5.4621e-03 0.0023 0.0037 0.0060
DWLS 5.4621e-03 0.0020 0.0210 0.0231
BWLS 5.4621e-03 0.0017 0.0088 0.0105
8 8
FDM 3.4173e-03 5.3493 0.0010 5.3503
HWLS 3.4173e-03 0.0566 0.0195 0.0761
DWLS 3.4173e-03 0.0050 0.1051 0.1101
BWLS 3.4173e-03 0.0038 0.0162 0.0200
16 16
FDM 1.9539e-03 4.6853 0.0020 4.6873
HWLS 1.9539e-03 0.0023 0.0037 0.0060
DWLS 1.9539e-03 0.0020 0.0205 0.0225
BWLS 1.9539e-03 0.0018 0.0088 0.0106
3232
FDM 1.0652e-03 4.3697 0.0029 4.3726
HWLS 1.0652e-03 0.0023 0.0037 0.0060
DWLS 1.0652e-03 0.0020 0.0205 0.0225
BWLS 1.0652e-03 0.0018 0.0088 0.0106
5. CONCLUSIONS
In this paper, we applied the Lifting schemes for the numerical solution of parabolic partial differential equations using different wavelet filters. The above figures shows that the
numerical solutions obtained by different Lifting schemes are agrees with the exact solution.
Also in the table, convergence of the presented schemes is observed i.e. the error decreases when the level of resolution N increases. In addition the calculations involved in Lifting schemes are simple, straight forward and low computation cost compared to classical method i.e. FDM. Hence the presented Lifting schemes in particular HWLS & BWLS are very effective for solving partial differential equations.
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