Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation

JOURNAL OF COMPUTATIONAL PHYSICS

卷:334

Article

Benner, Peter¹  Dolgov, Sergey²  Khoromskaia, Venera^1,3  Khoromskij, Boris N.³ 

[1] Max Planck Inst Dynam Complex Tech Syst, Sandtorstr 1, D-39106 Magdeburg, Germany

[2] Univ Bath, The Ave, Bath BA2 7AY, Avon, England

[3] Max Planck Inst Math Sci, Inselstr 22-26, D-04103 Leipzig, Germany

关键词: Bethe-Salpeter equation;    Hartree-Fock calculus;    Tensor decompositions;    Quantized-TT format;    Model reduction;    Structured eigensolvers;    Low-rank matrix;   

DOI  :  10.1016/j.jcp.2016.12.047

来源: Elsevier

【 摘 要 】

In this paper, we propose and study two approaches to approximate the solution of the Bethe-Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe-Salpeter operator. The complexity is reduced to O(N-b(2)) in the size of the atomic orbitals basis set, N-b, instead of the practically intractable 0(N-b(2)) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, No < Nb, hence the overall asymptotic complexity for solving the BSE problem by the QTT approximation is estimated by 0(log(N-0)N-o(2)). We confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chain-type molecules, while supporting sufficient accuracy. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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