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Sparse grids for the Schrödinger equation

Published online by Cambridge University Press:  16 June 2007

Michael Griebel  and
Jan Hamaekers
Michael Griebel
Affiliation:
Institute of Numerical Simulation, University of Bonn, Wegelerstraße 6, 53115 Bonn, Germany. griebel@ins.uni-bonn.de; hamaekers@ins.uni-bonn.de
Jan Hamaekers
Affiliation:
Institute of Numerical Simulation, University of Bonn, Wegelerstraße 6, 53115 Bonn, Germany. griebel@ins.uni-bonn.de; hamaekers@ins.uni-bonn.de
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Abstract

We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equation and compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system.


Keywords

Schrödinger equationnumerical approximationsparse grid methodantisymmetric sparse grids.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 2: Special issue on Molecular Modelling , March 2007 , pp. 215 - 247
Copyright
© EDP Sciences, SMAI, 2007

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