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

TL;DR

This paper introduces a fast, low-cost iterative method for solving the large-scale Bethe-Salpeter eigenvalue problem using low-rank and tensor train approximations, significantly reducing computational complexity while maintaining accuracy.

Contribution

The paper develops a novel iterative eigensolver leveraging low-rank and QTT tensor approximations, reducing computational costs from d7O(N_b^6) to d7O(\,log(N_o) N_o^2) for large-scale BSE problems.

Findings

Significant reduction in computational time demonstrated on molecular systems.

Enhanced accuracy with controlled numerical cost using reduced-block approximation.

Asymptotic complexity estimated as d7O(\,log(N_o) N_o^2).

Abstract

In this paper, we study and implement the structural iterative eigensolvers for the large-scale eigenvalue problem in the Bethe-Salpeter equation (BSE) based on the reduced basis approach via low-rank factorizations in generating matrices, introduced in the previous paper. The approach reduces numerical costs down to $\mathcal{O}(N_b^2)$ in the size of atomic orbitals basis set, $N_b$, instead of practically intractable $\mathcal{O}(N_b^6)$ complexity scaling for the direct diagonalization of the BSE matrix. As an alternative to rank approximation of the static screen interaction part of the BSE matrix, we propose to restrict it to a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate that the enhanced reduced-block approximation exhibits higher precision within the controlled numerical cost, providing as well a distinct two-sided error estimate for the BSE eigenvalues. It is shown that further reduction of the asymptotic computational cost is possible due to ALS-type iteration in block tensor train (TT) format applied to the quantized-TT (QTT) tensor representation of both long eigenvectors and rank-structured matrix blocks. The QTT-rank of these entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, $N_o$, hence the overall asymptotic complexity for solving the BSE problem can be estimated by $\mathcal{O}(\log(N_o) N_o^{2})$. We confirm numerically a considerable decrease in computational time for the presented iterative approach applied to various compact and chain-type molecules, while supporting sufficient accuracy.