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Tensor numerical methods in quantum chemistry

Author: Venera Khoromskaia; Boris N Khoromskij
Publisher: Berlin ; Boston : De Gruyter, [2018] ©2018
Edition/Format:   eBook : Document : Government publication : EnglishView all editions and formats
Summary:
The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Print version:
Material Type: Document, Government publication, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Venera Khoromskaia; Boris N Khoromskij
ISBN: 9783110365832 3110365839 9783110370157 3110370158
OCLC Number: 1041994386
Language Note: In English.
Description: 1 online resource (297 pages)
Contents: Frontmatter --
Contents --
1. Introduction --
2. Rank-structured formats for multidimensional tensors --
3. Rank-structured grid-based representations of functions in ℝd --
4. Multiplicative tensor formats in ℝd --
5. Multidimensional tensor-product convolution --
6. Tensor decomposition for analytic potentials --
7. The Hartree-Fock equation --
8. Multilevel grid-based tensor-structured HF solver --
9. Grid-based core Hamiltonian --
10. Tensor factorization of grid-based two-electron integrals --
11. Fast grid-based Hartree-Fock solver by factorized TEI --
12. Calculation of excitation energies of molecules --
13. Density of states for a class of rank-structured matrices --
14. Tensor-based summation of long-range potentials on finite 3D lattices --
15. Range-separated tensor format for many-particle systems --
Bibliography --
Index
Responsibility: Venera Khoromskaia, Boris N. Khoromskij.

Abstract:

When applied to multidimensional problems, conventional numerical methods suffer from the so-called "curse of dimensionality", which cannot be eliminated by parallel methods and high-performance  Read more...

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Khoromskij<\/span>\n\u00A0\u00A0\u00A0\nschema:author<\/a> <http:\/\/experiment.worldcat.org\/entity\/work\/data\/5276679898#Person\/khoromskaia_venera<\/a>> ; # Venera Khoromskaia<\/span>\n\u00A0\u00A0\u00A0\nschema:bookFormat<\/a> schema:EBook<\/a> ;\u00A0\u00A0\u00A0\nschema:datePublished<\/a> \"2018<\/span>\" ;\u00A0\u00A0\u00A0\nschema:description<\/a> \"The conventional numerical methods when applied to multidimensional problems suffer from the so-called \"curse of dimensionality\", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a \"smart\" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 1015. Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects. The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing. 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