利用低秩和QTT张量近似快速求解贝塔-萨尔佩特特征值问题
Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation
AI总结：
本文提出基于降阶方法的结构迭代求解器，用于求解大规模贝塔-萨尔佩特方程的特征值问题，通过低秩分解减少计算复杂度，同时利用QTT张量近似提高精度和效率。
摘要：
本文研究并实现了基于降阶方法的结构迭代求解器，用于求解大规模贝塔-萨尔佩特方程的特征值问题。通过在生成矩阵中使用低秩分解，该方法将数值成本降低到原子轨道基组大小$N_b$的$\mathcal{O}(N_b^2)$，而非直接对角化BSE矩阵的$\mathcal{O}(N_b^6)$复杂度。作为替代方案，我们限制静态屏互动部分到一个小的活跃子块，其大小平衡了其他矩阵块的秩结构表示的存储需求。我们证明增强的降阶块近似在受控的数值成本下具有更高的精度，并提供了BSE特征值的双侧误差估计。进一步减少渐近计算成本的可能性是通过应用块张量列车（TT）格式中的ALS型迭代，用于量化TT（QTT）张量表示的长特征向量和秩结构化矩阵块。这些实体的QTT秩几乎与分子系统中占据的轨道数$N_o$相同，因此求解BSE问题的总体渐近复杂度可估计为$\mathcal{O}(\log(N_o) N_o^{2})$。我们通过数值验证了所提出的迭代方法在各种紧凑型和链型分子上的计算时间显著减少，同时保持足够的准确性。
英文摘要：
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.
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