he present section is a necessary
prerequisite for the section (
Sparse
tensor product
). The reference is
[Walnut]
.
Wavelets are a preferable way to construct approximation spaces when applying
finite element method to PDEs. Decomposition with respect to wavelet basis has
a fast implementation similar to fast Fourier transform. Wavelets replicate
polynomials and have efficiency of decomposition comparable to Gauss-Hermite
integration. Wavelet decompositions have natural and stable subspace
splittings and thus allow for efficient preconditioners suitable for parallel
calculations. Wavelets form bases suitable for sparse tensor product-based
representation and thus do not have the curse of dimensionality.
The idea of wavelets may be illustrated by considering an attempt to
approximate generic functions with elementary shapes. We introduce the mesh
and define a constant function on every interval
:
As we increase the scale
we get finer approximation. Note that
is obtained by scaling and translation from the elementary shape
The closures of linears spans
form and increasing sequence of
spaces
The wavelets arise when we decide not to discard information while going from
to
.
Instead, we would like to produce a basis of the increment space
:
We would like such basis to have the form
for some function
.
In addition, we may want to increase complexity of the elementary shape
so
that
would include polynomials up to some degree. In addition, we want supports of
and
to be finite and minimal. We also may want to have symmetry of some kind. Then
we might want to restrict such construction to an interval and force it to
satisfy boundary conditions. Finally, we need such construction to have strong
stability with respect to subspace decompositions.
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