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. Wavelets replicate polynomials and thus have
efficiency of approximation. Wavelet decompositions have natural and stable
subspace splittings and thus allow for preconditioners suitable for parallel
calculations. Wavelets form bases suitable for sparse tensor product-based
representation. Such bases grow conservatively when increasing dimensionality.
Multiscale structure of wavelet bases is naturally suitable for construction
of adaptive grids.
The idea of wavelets may be illustrated by considering an attempt to
approximate generic functions with elementary shapes. We introduce a 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 linear 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
:
Such structure is essential when constructing adaptive grid.
We would like such basis to have the form
span
and
span
for some functions
and
.
In addition, we may want to increase complexity of the elementary shape
so
that
would include polynomials up to some degree. Moreover, 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. It is remarkable that we
can have it all. The following sections contain detailed derivations and
occasional Mathematica scripts. The part
(
Numerical analysis part
) contains some
practical Python scripts and C++/Cuda codes.
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