Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation
Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation
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DOI：
10.1016/j.jcp.2016.12.047
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发表时间：
2016-02
期刊：
影响因子：
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通讯作者：
P. Benner;S. Dolgov;V. Khoromskaia;B. Khoromskij
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文献类型：
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作者：
P. Benner;S. Dolgov;V. Khoromskaia;B. Khoromskij
在本文中，我们提出并研究了两种使用结构化迭代特征值求解器来近似求解 Bethe-Salpeter 方程 (BSE) 的方法。两种方法都基于简化基方法和生成矩阵的低秩因式分解。我们还建议通过一个小的活动子块来表示 BSE 矩阵中的静态屏幕交互部分，其大小平衡其他矩阵块的排序结构表示的存储。我们通过各种数值测试证明，对角线加低秩加缩减块近似的组合表现出较高的精度和较低的数值成本，并为 Bethe-Salpeter 算子的最小特征值提供了独特的两侧误差估计。原子轨道基组 N b 的大小将复杂性降低至 O (N b 2)，而不是直接对角化中实际难以处理的 O (N b 6) 缩放。在第二种方法中，我们将量化 TT (QTT) 张量表示应用于秩结构 BSE 矩阵块中的长特征向量和列向量，并将其与块 QTT 格式中的 ALS 类型迭代相结合。矩阵实体的 QTT 秩几乎与分子系统中占据轨道的数量相同，N o< N b，因此通过 QTT 近似求解 BSE 问题的整体渐近复杂度由 O (log(N o) N o 2) 估计。我们在数值上证实，应用于各种紧凑型和链型分子的迭代方法的计算时间显着减少，同时支持足够的精度。
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 O (N b 6) 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, N o< N 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 chain-type molecules, while supporting sufficient accuracy.