Phase I Final Report:
Parallel Methods
for
the Schroedinger Equation
using
Domain Decomposition
and
the Wavelet-Capacitance Matrix. ¹
U.S. Department of Commerce
SBIR Grant, 50-DKNA-4-0099

John Weiss
Applied Mathematics Group
49 Grand View Road
Arlington, MA 02476

January 20, 1995

Abstract:

We have developed a parallel domain decomposition method based on the eigenstates of the capacitance matrix for domains with symmetry. A complete description of the orthogonal subspace structures has been found, leading to efficient algorithms for the calculation of the eigenstates. The use of symmetry simplifies the matching conditions along the boundaries of elements. We define an adaptive algorithm by a natural expansion of basis elements by basis elements at a smaller scale. An extension of these results to three dimensions is found by a connection of symmetry with the crystallographic group of transformations.

We have developed parallel, wavelet-Galerkin methods for the Schroedinger equations in two space dimensions with singular, particle potentials. A linear speedup was found. One conclusion of this work is that : A wavelet-Galerkin algorithm for the Schroedinger equations with practically, complete linear speedup is implemented. The only limitation on the speed of the algorithm is the internode communication required for the FFT. Therefore, given a distributed system with a large number of nodes, the speed and efficiency of the Schroedinger wavelet-Galerkin solvers are asymptotic to the speed and efficiency of the 2D-FFT defined on a domain decomposition that itself will scale with the number of nodes. For this reason, we have developed optimized two-dimensional, parallel FFT algorithms for use with the wavelet-Galerkin Schroedinger equation solvers.