Solving large-scale eigenvalue problems poses a significant challenge due to the computational complexity and limitations on the parallel scalability of the orthogonalization operation, when many eigenpairs are required. In this paper, we propose an intrinsic orthogonality-preserving model, formulated as an evolution equation, and a corresponding numerical method for eigenvalue problems. The proposed approach automatically preserves orthogonality and exhibits energy dissipation during both time evolution and numerical iterations, provided that the initial data are orthogonal, thus offering an accurate and efficient approximation for the large-scale eigenvalue problems with orthogonality constraints. Furthermore, we rigorously prove the convergence of the scheme without the time step size restrictions imposed by the CFL conditions. Numerical experiments not only corroborate the validity of our theoretical analyses but also demonstrate the remarkably high efficiency of the algorithm.

翻译：求解大规模特征值问题面临重大挑战，原因在于计算复杂度高，且当需要大量特征对时，正交化操作的并行可扩展性存在局限。本文提出一种内禀保持正交性的模型，该模型以演化方程形式构建，并给出相应的特征值问题数值方法。所提方法在初始数据正交的前提下，能自动保持正交性，并在时间演化和数值迭代过程中呈现能量耗散特性，从而为具有正交约束的大规模特征值问题提供精确高效的近似解。此外，我们严格证明了该格式的收敛性，且无需受CFL条件限制的时间步长约束。数值实验不仅验证了理论分析的正确性，还证明了该算法具有显著的高效性。