Abstract:
Atomic decompositions of functions are a powerful analysis tool but
also
provide a link to the practice of numerical discretization methods. While
some special cases such as wavelet decompositions or multiscale methods
with finite elements nodal basis functions are well-understood by now,
others still pose a challenge, and invite with open problems, both on the
theoretical and the practical side.
The talk will concentrate on so-called multilevel partition of unity
methods (MPUM) which have emerged from the generalized finite element
method and hp-methods (Babuska, Oden, Belychko, Liu, Griebel, et al.) and
became popular in computational engineering, on the one hand, and
quarkonial decompositions introduced by Triebel in the late 90-ies to
study local and global properties of functions on Rn, manifolds, fractals,
etc., on the other. Both are based on locally supported "atoms" of the
form \pi j \lambda (x) = \pi j \lambda (x)(x- \lambda)^\beta
where the (equation) are smooth window functions supported in balls of
radius (equation) and centers (equation) which form
-equation-
a partition of unity on each level (equation). Roughly speaking, the
point set (equation) resembles a perturbation of a lattice with mesh-width
(equation). Compared to other multiscale systems, the additional
ingredient is the parameter (equation) which allows for local Taylor
decomposition. This leads to highly redundant systems which are also
connected wih hp-discretization methods where local grid refinement is
combined with higher-degree polynomial approximation.
The talk will be introductory, and rather state some unsolved problems
than present grand new results (we don't have them yet!).
|
