Inhalt des Dokuments
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)
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.