Error Estimation and Assessment of an Approximation in a Wavelet Collocation Method ()
1. Introduction
In the wavelet theory a scaling function 
is used, which has properties that are defined in the MSA (multi scale analysis). Through the MSA we know, we can construct an orthonormal basis of a closed subspace
, where 
belongs to a sequence of subspaces with the following property:
is an orthonormal basis of 
with
.
We use the following approximation function:
, with
.
and 
depend on the approximation interval
.
Now we can approximate the solution of an initial value problem 
and 
by minimizing the following function (
is the Euklid norm)
(1)
For 
we get an equivalent problem:
, with 
and
.
The advantage of calculating 
by minimizing 
is that we can choose more collocation points 
as shown in the following example. In that case we apply the least squares method to calculate
. Many simulations had shown that if 
was very small then the approximation yj would be good. An even better criterion for a good approximation 
is 
(see (3)). Moreover, the equations have been ill-conditioned in several examples.
Analogously we could use boundary conditions instead of the initial conditions. This method can be even used analogously for PDEs, ODEs of higher order or DAEs, which have the form
If 
is an ODE system, then we use the approximation function:
For the i-th component of the solution y, we use the notation 
as usual. We use for the i-th component of 
the notation
, in order not to lead to a confusion with the approximation 
out of
, so it will be always distinguished whether the approximation 
or the i-th component of 
is used.
We use the collocation points
, with 
and
(2)
Simulations have shown that even with 
we get good approximations.
For the assessment of the approximation we use the value
, with
(3)
, 
and 
is an integer. For big 
we should weight 
with
.
Remarks 1:
1) We get
for
, because of:
Analogously for smaller
.
2) The sums in (1) and (3) could start with
, too.
3) 
in (1) could also be used as a constraint if the initial value should be fulfilled. But in all good approximations, 
was very small.
In the examples we use the Shannon wavelet. Although it has no compact support and no high order, in many examples and simulations we got a much better approximation than using other wavelets (f.e. Daubechies wavelets of order 5 to 8), even with a small n. The Meyer wavelet yields good results, too.
We even get a good extrapolation outside the interval
.
Example 1:
1) We use the following ODE
The exact solution is
.
We approximated the solution on the interval 
and chose
, like in all examples.
With 
we could see in all our simulations, if the approximation was good. We got a linear relationship between 
and
. In Figure 1 we see the graph of a linear regression (with an R squared of 0.991196) of 
against 
with the points
, which have been calculated with different
and 
with the ODE and I of the example 1.
is the mean squared error
with
.
Now we see a regression table (Table 1) of 
on
, which shows a linear dependency in our example and the graph of the linear regression function.
Here is a graph of the regression function and the graphs of the functions yi and 
for j = 0, kmax = 15 and 
on the approximation interval 
(see Figures 2 and 3) and on the interval 
(see Figures 4 and 5). In Figures 4 and 5 we see that we get even a good extrapolation.
Figure 1. Linear regression plot of 
against
.
Table 1. Linear regression table of 
on
.
2. Error Estimation and Assessment of the Approximation
In the example we used the Shannon wavelet. For this wavelet we have additional information about the error in the Fourier space from the Shannon theorem. For a good approximation with a small j the behavior of 
with growing 
is important, because (if yi is an orthogonal projection from y on Vj and
)
With the Parseval theorem we get
so
With the Riemann-Lebesgue theorem we get:
For the approximation error the decay behaviour of the detail coefficients 
is important:
On the other side: we have got in many simulations with the Shannon wavelet better approximations (with the described collocation method) than with higher order wavelets.
Remarks 2:
1) For a theoretical multi resolution analysis we could consider 
instead of
, because when 
is in 
then 
is in
, if we need an approximation on
. Here 
is the indicator function of the interval
.
2) For interpolating wavelets there are a number of publications with error estimates and also for the approximation of the solutions of initial value problems and boundary value problems (for ordinary and partial differential equations) see [1,2], as well as to the sinc collocation method (see [3-5]) with special collocation points (“sinc grid points”, see [5]).
Theorem 1 (for the decay behaviour):
The wavelet 
has the order
, 
with 
and 
is Lipschitz continuous. Then exists a 
independent from 
with
is the wavelet transform of 
with
A proof is in [6]. So we get for the detail coefficients an appraisal because
and so
Now we saw that the decay of the detail coefficients depends on the order of a wavelet.
From the Gilbert-Strang Theory (see [7]) we know additionally an upper bound of the approximation error in dependency of the order
: if the wavelet is of order 
then the approximation error has the order 
if 
and (if 
is an orthogonal projection from 
on 
and
)
.
If a wavelet is of order 
the scaling function 
even has an interpolation property, because then we can construct the functions 
with 
over a linear combination of 
(see [7]). That’s also a property of the so called interpolating wavelets. For interpolating wavelets we find error estimations in [8] and [9].
Remarks 3:
1) Error estimations for the sinc collocation with a transformation can be found in [4] and [5].
2) Although the approximation error is depended on the order of a wavelet in many simulations the Shannon wavelet led to much better approximations than Daubechies wavelets of higher order, if the approximation function 
was calculated by minimizing the sum of squares of residuals Q. Even when comparing the extrapolations the Shannon wavelet was significantly better.
The reason is, that we do not calculate an orthogonal projection on 
like in the appraisal above and the function y is in general case not quadratic integrabel on R (we consider only a compact interval I).
The following appraisal takes account of the fact that we calculate the approximation function by the minimization of Q. We first need a theorem, which follows from the Gronwall-Lemma.
Theorem 2:
Assumptions: we have a initial value problem 
with 
and
(4)
and
(5)
Then we get for
:
For a proof see [10].
Theorem 3:
With the assumptions from Theorem 2 we get (if 
):
So we get the follow inequality for
, which is used in the example 2:
(6)
depends on 
and 
only on the initial value problem and the collocation points. We write 
instead of 
because in example 2 we set 
on the x-axes so we have a comparison with example 1 where we set 
on the x-axes.
Remark 4:
We get with 
If additionally 
for one (or more) 
we get:
This is analogously right for 
instead if 
with
and 
and an integer
. Qa is an upper bound for
. With 
we could assess in all simulations the quality of an approximation and in linear regressions from 
on 
we got in almost all simulations a 
(R squared) greater than 0.99 (see next example). Only if all approximations have been bad, then 
was less than 0.99 (but we still have a dependency). If 
is the exact solution, then
. Because we get not only a approximation with points (we get a approximation function
) we must not calculate a second minimization for the calculation of
.
will be in general (for
) less than M, because we use the collocations points ti and so 
is very small at these points (see the next graphic). 
was in many good simulations less than 10−16.
In many simulation 
is relative big between to collocation points (or at the edge of I if we start with i = 1 in the sum (1)).
In Figure 6 we see the graph of
in example 1 for 
and
. Here a too small 
results in a very bad approximation.
We see that 
could be very small with a too small m, but 
is very big here. In the graph we see that d is very small at the collocation points 
but between them d is very big. That’s the reason because we could identify with 
a worse approximation in any our simulations. On the other hand a big 
is an indicative of a too small j.
So we can approximate M here with the maximum of 
at the points 
with 
like we do it in the next example.
Now we want to apply the result from theorem 2. Furthermore we will see a correlation between an approximation of 
and 
in this example like we saw it before between 
and
.
Example 2:
We use the initial value problem and the approximations with the different parameters
, 
and m of example 1. If 
than follows from theorem 2 (under the assumptions from this theorem):
Here we get (see (6)):
We now apply a linear regression of 
on 
with the approximation
from M2 with 
(the points from 
beginning with
). sse and 
have been calculated with the points 
(and the summation indices
).
Here is the regression table (Table 2) (with a R squared of 0.986877).
In Figure 7 we see a graph from 
(in red), the graph of the regression function (in blue) and the regression points
. 
was not considered (this means we set
) because it was very small.
Here are the graphs of 
with
, 
and 
in Figure 8. In most simulations 
was less than 10−16.
Generally we can use
,
and 
(with an integer
) for an approximation of
. Here we know the following relation:
3. Conclusions
We defined a variable 
with which you can evaluate an approximation. In many simulations and in the examples of this article we saw that we get good results with
. A linear relationship between 
and 
was shown in example 1. It is also shown that the approximation can be used to extrapolate outside the approximation interval.
Using Theorem 2 we derive an estimate (see theorem 3). Then it is shown how to detect a too great step size using
. In example 2 we show that the deduced estimate represents a straight line (in the coordinate system with 
on the x-axes and 
on the yaxes), which runs approximately parallel to the regression line (it is approximately parallel because the regression function is an estimation, theoretically it must be parallel because it cannot cross the upper bound line). In a research project we got analogous results in many
Figure 7. Linear regression plot of 
against
.
simulations, even with systems and higher order odes.
It is shown that 
(the size of the estimate) can be approximated via
, and this approximation has 
as upper bound. The regression of the points
returns a slightly larger 
than the regression with the points
. As a consequence, Q2 is well suited to assess, especially as you can estimate the approximation of 
with Q2 and in Q2 more information is included. Moreover we can compare Q2 with 
to assess the approximation (see Figure 6).