In this project, we propose to use a sparse grid method for the
direct discretization of Schrödinger's equation. The conventional
sparse grid technique allows to reduce the complexity of a
d-dimensional problem from 
to
,
provided that certain smoothness assumptions are fulfilled. It
uses a multi-level basis to represent one-particle states and
employs a certain determinant-product approach to represent
many-particle states, which takes anti-symmetry (Pauli principle)
into account.
A certain truncation of the corresponding multi-level series
expansion directly results in a cost-optimal discretization of the
total electronic space. Here, a dimension-adaptive procedure
allows to detect correlations between one-particle states. This
new approach gives the perspective to reduce the computational
complexity of a 
-electron problem to that of a one-electron
problem.
For different choices of multi-level bases (real space, Fourier space) for the one-particle state, we will implement the resulting dimension-adaptive sparse grid approaches and compare their properties for Schrödinger's equation. Furthermore, this code will later be parallelized and implemented on distributed memory processors.
| [1] | J. Garcke and M. Griebel. On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. Journal of Computational Physics, 165(2):694-716, 2000. also as SFB 256 Preprint 670, Institut für Angewandte Mathematik, Universität Bonn, 2000. |
| [2] | J. Garcke. Berechnung von Eigenwerten der stationären Schrödingergleichung mit der Kombinationstechnik Diplomarbeit, Institut für Angewandte Mathematik, Universität Bonn, 1998. |