Geodesic
Sparse grids for the Schrödinger equation
ESAIM: Mathematical Modelling and Numerical Analysis , Special issue on Molecular Modelling, Tome 41 (2007) no. 2, pp. 215-247

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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.

DOI : 10.1051/m2an:2007015
Classification : 35J10, 65N25, 65N30, 65T40, 65Z05
Keywords: Schrödinger equation, numerical approximation, sparse grid method, antisymmetric sparse grids 
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     author = {Griebel, Michael and Hamaekers, Jan},
     title = {Sparse grids for the {Schr\"odinger} equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {215--247},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {2},
     year = {2007},
     doi = {10.1051/m2an:2007015},
     mrnumber = {2339626},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2007015/}
}
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Griebel, Michael; Hamaekers, Jan. Sparse grids for the Schrödinger equation. ESAIM: Mathematical Modelling and Numerical Analysis , Special issue on Molecular Modelling, Tome 41 (2007) no. 2, pp. 215-247. doi: 10.1051/m2an:2007015

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