Sparse grids for the Schrödinger equation

Michael Griebel; Jan Hamaekers

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 2, page 215-247
  • ISSN: 0764-583X

Abstract

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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.


How to cite

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Griebel, Michael, and Hamaekers, Jan. "Sparse grids for the Schrödinger equation." ESAIM: Mathematical Modelling and Numerical Analysis 41.2 (2007): 215-247. <http://eudml.org/doc/250099>.

@article{Griebel2007,
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.
},
author = {Griebel, Michael, Hamaekers, Jan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Schrödinger equation; numerical approximation; sparse grid method; antisymmetric sparse grids.; space grid method; anti-symmetric sparse grids; many-particle problems; fermionic systems; electronic Schrödinger equation; convergence; scalability},
language = {eng},
month = {6},
number = {2},
pages = {215-247},
publisher = {EDP Sciences},
title = {Sparse grids for the Schrödinger equation},
url = {http://eudml.org/doc/250099},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Griebel, Michael
AU - Hamaekers, Jan
TI - Sparse grids for the Schrödinger equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/6//
PB - EDP Sciences
VL - 41
IS - 2
SP - 215
EP - 247
AB - 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.

LA - eng
KW - Schrödinger equation; numerical approximation; sparse grid method; antisymmetric sparse grids.; space grid method; anti-symmetric sparse grids; many-particle problems; fermionic systems; electronic Schrödinger equation; convergence; scalability
UR - http://eudml.org/doc/250099
ER -

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