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## Abstracts

**Randolph E. Bank***On the
Convergence of Adaptive Feedback Loops*

We present a
technique for proving convergence of h and hp adaptive finite element
methods through comparison with certain reference refinement schemes
based on interpolation error. We then construct a testing environment
where properties of different adaptive approaches can be evaluated and
improved.

This is a joint-work with Professor Dr. Harry
Yserentant (TU Berlin)

**Michael Griebel***An adaptive many body
expansion approach for efficient electronic structure
calculations*

After discussing the specific bounded mixed regularity of the
solution of the electronic Schrödinger equation due to Yserentant, we
present a multi–scale decomposition approach for its efficient
approximate ground state calculation. It is based on an
Anova-like dimension-wise decomposition of the solution space and
represents the energy of atoms and molecules as a finite sum of
contributions which depend on the positions of single nuclei, of pairs
of nuclei, of triples of nuclei, and so on. Under the assumption
of locality of the electronic wave functions, the higher order terms
in this expansion decay rapidly and may therefore be truncated.
This way, only the calculation of the electronic structure of local
parts, i.e. of small subsystems of size *k* of the overall
system, is necessary to approximate the total ground state
energy.

This decomposition approach is combined with the number *p*
of approximation functions in the discretization of the
*k*-sized subsystems. Then, it turns out that a sparse
grid-like approach in the parameters *k* and *p* results
in a very fast and parallel solution procedure which allows the
numerical treatment of huge bio-molecules in decent run time and
results in excellent approximations.

Our approach can be used in an adaptive sparse-grid fashion which speeds up run times even further.