发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences; City University of Hong Kong; Guangdong University of Technology（中国科学院数学与系统科学研究院; 中国科学院大学; 香港城市大学; 广东工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高维极端特征值问题，本文提出基于张量列格式的低秩参数化与流形优化方法，显著降低计算成本，并在多个数值实验中达到全空间求解器精度且具备鲁棒性。
AI 中文摘要
高维极端特征值问题通常出现在分子振动模型、电子结构计算和量子力学中。直接求解这些问题会受到维度灾难的影响。我们不在高维环境空间中处理问题，而是通过低秩张量格式重新表述问题，这可以显著降低计算成本和存储需求。具体而言，我们考虑张量列格式中有界秩张量上的Rayleigh-Ritz问题，这允许更灵活地选择秩参数。此外，我们通过引入松弛变量来考虑有界秩张量的光滑参数化，从而形成光滑流形结构。因此，原始的Rayleigh-Ritz问题被转化为流形上的优化问题。我们发展了黎曼几何，并提出了求解极端特征值问题的优化方法。在实践中，利用哈密顿量和偏微分方程算子中的Kronecker积结构来简化计算。在谐振子、拉普拉斯算子、薛定谔方程、层状团簇问题和玻色-爱因斯坦模型上的数值实验表明，所提出的方法在减少计算时间的同时，达到了与全空间特征值求解器相当的精度，并避免了存储全向量。结果还显示了迭代过程中数值秩的降低，以及当秩参数被过度估计时的鲁棒性。此外，不动点迭代表明，所提出的方法能够作为求解非线性特征值问题的内部特征值求解器。
英文摘要
High-dimensional extreme eigenvalue problems often arise from molecular vibrational models, electronic structure calculations, and quantum mechanics. Directly solving these problems suffers from the curse of dimensionality. Instead of tackling the problem in high-dimensional ambient space, we reformulate the problem through low-rank tensor formats, which can significantly reduce the computational cost and storage. Specifically, we consider the Rayleigh--Ritz problem on bounded-rank tensors in the tensor train format, which enables a more flexible choice of rank parameters. Moreover, we consider a smooth parametrization for bounded-rank tensors by introducing slack variables, leading to a smooth manifold structure. The original Rayleigh--Ritz problem is therefore transferred to an optimization problem on the manifold. We develop the Riemannian geometry and propose optimization methods for solving extreme eigenvalue problems. In practice, the Kronecker-product structure in Hamiltonian and PDE operators is employed to simplify the computation. Numerical experiments on the harmonic oscillator, the Laplace operator, the Schrödinger equation, the layered cluster problem, and the Bose--Einstein model demonstrate that the proposed method achieves accuracy comparable to full-space eigensolvers with reduced computation time and avoids storing full vectors. The results also show numerical rank reduction during iteration and robustness when the rank parameter is over-estimated. In addition, the fixed-point iterations illustrate that the proposed methods are able to serve as an inner eigensolver for solving nonlinear eigenvalue problems.
Comments28 pages, 14 figures, 2 tables
