A quasi-Grassmannian gradient flow model for eigenvalue problems

TL;DR

This paper introduces a quasi-Grassmannian gradient flow model for eigenvalue problems that naturally ensures orthogonality, converges exponentially, and improves robustness without explicit orthogonalization.

Contribution

The paper presents a novel gradient flow model that inherently maintains orthogonality and guarantees exponential convergence for eigenvalue problems.

Findings

Model ensures asymptotic orthogonality without initial orthogonality.

Solutions converge exponentially to eigenvalues and eigenvectors.

Enhanced robustness compared to traditional methods.

Abstract

We propose a quasi-Grassmannian gradient flow model for eigenvalue problems of linear operators, aiming to efficiently address many eigenpairs. Our model inherently ensures asymptotic orthogonality: without the need for initial orthogonality, the solution naturally evolves toward being orthogonal over time. We establish the well-posedness of the model, and provide the analytical representation of solutions. Through asymptotic analysis, we show that the gradient converges exponentially to zero and that the energy decreases exponentially to its minimum. This implies that the solution of the quasi-Grassmannian gradient flow model converges to the solution of the eigenvalue problems as time progresses. These properties not only eliminate the need for explicit orthogonalization in numerical computation but also significantly enhance robustness of the model, rendering it far more resilient to numerical perturbations than conventional methods.