An orthogonality-preserving approach for eigenvalue problems

TL;DR

This paper introduces an orthogonality-preserving evolution equation and numerical method for large-scale eigenvalue problems, ensuring automatic orthogonality, energy dissipation, and convergence without CFL restrictions, leading to high efficiency.

Contribution

It presents a novel intrinsic orthogonality-preserving model and numerical scheme that automatically maintains orthogonality and converges efficiently without CFL constraints.

Findings

The method automatically preserves orthogonality during evolution.

The scheme exhibits energy dissipation and convergence without CFL restrictions.

Numerical experiments demonstrate high efficiency and validity.

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.