A common approach
for numerical methods is to describe the numerical solution as a linear combination
of basis functions. These should allow for accurate approximations to
the true solution by means of a
small number of actually used basis functions
. Wavelets (or multiscale basisfunctions) do! Two particular univariate examples of such multiscale bases are the Haar-wavelets (left) and the Hierarchical basis (right) below:
Fig. 1 wavelets of the first three levels for the so-called Haar- and the Hierarchical basis Multiscale functions are
denoted by
. Here The main reason for the approximation power of multiscale bases is that just a few basis functions from low (high) levels with large (small) support are sufficient to approximate the rather global smooth (localized non-smooth) parts of the target function. In the below example a function with steep gradients is approximated by a superposition of just four hat functions from the first four refinement levels.
The approximation power of the Haar- and Hierarchical Basis is rather weak. More favourable are high order generalizations of the hat functions: Interpolets [5]. They are smoother and they reproduce polynomials of higher degree. |