### Abstract

We propose new tensor approximation algorithms for certain discrete functions related with Hartree-Fock/Kohn-Sham equations. Given a canonical tensor representation for the electron density function (for example, produced by quantum chemistry packages such as MOLPRO), we obtain its Tucker approximation with much fewer parameters than the input data and the Tucker approximation for the cubic root of this function, which is part of the Kohn-Sham exchange operator. The key idea is in the fast and accurate prefiltering of possibly large-scale factors of the canonical tensor input. The new algorithms are based on the incomplete cross approximation method applied to matrices and tensors of order 3 and outperform other tools for the same purpose.First, we show that the cross approximation method is robust and much faster than the singular value decomposition-based approach. As a consequence, it becomes possible to increase the resolution of grid and the complexity of molecules that can be handled by the Hartree-Fock chemical models. Second, we propose a new fast approximation method for f^{1/3}(x, y, z), based on the factor prefiltering method for f(x, y, z) and certain mimic approximation hypothesis. Third, we conclude that the Tucker format has advantages in the storage and computation time compared with the ubiquitous canonical format.

Original language | English |
---|---|

Pages (from-to) | 935-952 |

Number of pages | 18 |

Journal | Numerical Linear Algebra with Applications |

Volume | 17 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Dec 2010 |

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### Keywords

- Canonical decomposition
- Cross approximation
- Electron density
- Hartree-Fock/Kohn-Sham
- Tucker approximation

### Cite this

*Numerical Linear Algebra with Applications*,

*17*(6), 935-952. https://doi.org/10.1002/nla.682

}

*Numerical Linear Algebra with Applications*, vol. 17, no. 6, pp. 935-952. https://doi.org/10.1002/nla.682

**Cross approximation in tensor electron density computations.** / Oseledets, I. V.; Savostyanov, D. V.; Tyrtyshnikov, E. E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Cross approximation in tensor electron density computations

AU - Oseledets, I. V.

AU - Savostyanov, D. V.

AU - Tyrtyshnikov, E. E.

PY - 2010/12/1

Y1 - 2010/12/1

N2 - We propose new tensor approximation algorithms for certain discrete functions related with Hartree-Fock/Kohn-Sham equations. Given a canonical tensor representation for the electron density function (for example, produced by quantum chemistry packages such as MOLPRO), we obtain its Tucker approximation with much fewer parameters than the input data and the Tucker approximation for the cubic root of this function, which is part of the Kohn-Sham exchange operator. The key idea is in the fast and accurate prefiltering of possibly large-scale factors of the canonical tensor input. The new algorithms are based on the incomplete cross approximation method applied to matrices and tensors of order 3 and outperform other tools for the same purpose.First, we show that the cross approximation method is robust and much faster than the singular value decomposition-based approach. As a consequence, it becomes possible to increase the resolution of grid and the complexity of molecules that can be handled by the Hartree-Fock chemical models. Second, we propose a new fast approximation method for f1/3(x, y, z), based on the factor prefiltering method for f(x, y, z) and certain mimic approximation hypothesis. Third, we conclude that the Tucker format has advantages in the storage and computation time compared with the ubiquitous canonical format.

AB - We propose new tensor approximation algorithms for certain discrete functions related with Hartree-Fock/Kohn-Sham equations. Given a canonical tensor representation for the electron density function (for example, produced by quantum chemistry packages such as MOLPRO), we obtain its Tucker approximation with much fewer parameters than the input data and the Tucker approximation for the cubic root of this function, which is part of the Kohn-Sham exchange operator. The key idea is in the fast and accurate prefiltering of possibly large-scale factors of the canonical tensor input. The new algorithms are based on the incomplete cross approximation method applied to matrices and tensors of order 3 and outperform other tools for the same purpose.First, we show that the cross approximation method is robust and much faster than the singular value decomposition-based approach. As a consequence, it becomes possible to increase the resolution of grid and the complexity of molecules that can be handled by the Hartree-Fock chemical models. Second, we propose a new fast approximation method for f1/3(x, y, z), based on the factor prefiltering method for f(x, y, z) and certain mimic approximation hypothesis. Third, we conclude that the Tucker format has advantages in the storage and computation time compared with the ubiquitous canonical format.

KW - Canonical decomposition

KW - Cross approximation

KW - Electron density

KW - Hartree-Fock/Kohn-Sham

KW - Tucker approximation

UR - http://www.scopus.com/inward/record.url?scp=78649667629&partnerID=8YFLogxK

U2 - 10.1002/nla.682

DO - 10.1002/nla.682

M3 - Article

AN - SCOPUS:78649667629

VL - 17

SP - 935

EP - 952

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 6

ER -