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

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 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(Nb2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O(Nb6) 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 O(log⁡(No)No2). 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.