scripts basis3.py, basis3a.py, project.py in the directory
OTSProjects/wavelets2 contain the following derivations.
We would like to obtain a decomposition for a
with respect to a wavelet basis adapted to the interval
We start from results of the procedure
of approximation spaces in one dimension II.
) We have the sets of
We adapt these functions to the interval
by applying the operation
normalize them to
This is done in basis3.py and basis3a.py.
In the script project.py we initially
is the minimal scale that satisfies the condition
Sufficiently fine scale 2
) and the
projection is taken in
We form the
for some fixed
Then we repeatedly
reaches a given number.
This experiment yields the following conclusions:
1. Approximation is significantly better then anything that may be obtained by
classical finite elements or low order wavelets.
2. Floating point errors break procedure for reasonably large
if we use the Poly class. We successfully remove the difficulty using the
localized piecewise polynomial representation of the functions
covered in the section
of localized piecewise polynomial functions
3. Accelerated version of the procedure involves taking several functions of
the same scale on the
The number of functions decreases with scale and increases with required
Another way to accelerate the procedure is to gradually increase the set
based on results of previous steps.
One does not need to calculate wavelet decomposition at every evaluation
because of scaling and translation properties of the wavelet basis. The
decomposition may assembled almost instantly and with perfect precision from
executed using LPoly.
The script dproject.py contains a similar experiment performed in
Note that the resulting recipe is opposite to the recipe of the section
Indeed, the wavelet basis approximates perfectly the linear part of the
and has difficulty around the singularity. However, it is around the
singularity where the function
has the smallest variation.