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Abstract
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 lowrank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active subblock, with a size balancing the storage for rankstructured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus lowrank plus reducedblock approximation exhibits higher precision with low numerical cost, providing as well a distinct twosided 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 O(N_{b} ^{6}) scaling for the direct diagonalization. In the second approach, we apply the quantizedTT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rankstructured BSE matrix blocks, and combine this with the ALStype iteration in block QTT format. The QTTrank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, N_{o}b, hence the overall asymptotic complexity for solving the BSE problem by the QTT approximation is estimated by O(log(N_{o})N_{o} ^{2}). We confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chaintype molecules, while supporting sufficient accuracy.
Original language  English 

Pages (fromto)  221239 
Number of pages  19 
Journal  Journal of Computational Physics 
Volume  334 
Early online date  5 Jan 2017 
DOIs  
Publication status  Published  1 Apr 2017 
Keywords
 Bethe–Salpeter equation
 Hartree–Fock calculus
 Lowrank matrix
 Model reduction
 QuantizedTT format
 Structured eigensolvers
 Tensor decompositions
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Projects
 2 Finished

Tensor product numerical methods for highdimensional problems in probability and quantum calculations
1/01/16 → 31/12/18
Project: Research council