An orthogonality-preserving approach for eigenvalue problems

Abstract

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.