一种保持结构的LOBPCG算法用于贝泰-萨尔佩特本征值问题
A Structure-Preserving LOBPCG Algorithm for the Bethe-Salpeter Eigenvalue Problem
AI总结：
本文提出了一种保持结构的LOBPCG算法，用于高效求解贝泰-萨尔佩特本征值问题，并展示了其在对称正定矩阵辛本征值问题中的应用潜力。
摘要：
贝泰-萨尔佩特本征值问题是一种出现在多体物理中的结构化本征值问题。在实践中，需要计算其中几个最小的正本征值及其对应的本征向量。原则上，LOBPCG算法可以用来解决这个问题。然而，现有LOBPCG算法的直接应用并未利用该问题固有的结构。我们设计了一种基于不定LOBPCG算法的结构保持本征解算器，以高效解决贝泰-萨尔佩特本征值问题。我们提出了一个改进的Hetmaniuk-Lehoucq技巧用于不定内积，以及一种自适应的多级正交化策略，以确保算法的数值稳定性。数值实验表明，所提出算法能够高效且准确地计算所需的本征对。由于对称正定矩阵的辛本征值问题可以转化为贝泰-萨尔佩特本征值问题，我们的算法可以自然地作为辛本征解算器被采用。
英文摘要：
The Bethe-Salpeter eigenvalue problem is a structured eigenvalue problem arising in many-body physics. In practice, a few of the smallest positive eigenvalues and the corresponding eigenvectors need to be computed. In principle, the LOBPCG algorithm can be applied to solve this eigenvalue problem. However, direct application of the existing LOBPCG algorithm does not utilize the inherent structure of the problem. We design a structure-preserving eigensolver based on the indefinite LOBPCG algorithm to efficiently solve the Bethe-Salpeter eigenvalue problem. We propose an improved Hetmaniuk-Lehoucq trick for the indefinite inner product, as well as an adaptive, multi-level orthogonalization strategy to ensure the numerical stability of our algorithm. Numerical experiments demonstrate that the proposed algorithm can efficiently and accurately compute the desired eigenpairs. Since the symplectic eigenvalue problem for symmetric positive definite matrices can be transformed to the Bethe-Salpeter eigenvalue problem, our algorithm can naturally be adopted as a symplectic eigensolver.
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