A quasi-Grassmannian gradient flow model for eigenvalue problems
This work provides a new continuous-flow framework for eigenvalue problems that relaxes the need for initial orthogonality, which may be useful for numerical methods in scientific computing.
The paper proposes a quasi-Grassmannian gradient flow model for eigenvalue problems that ensures asymptotic orthogonality without requiring initial orthogonality, and proves exponential convergence of the gradient and energy to zero and the minimum, respectively.
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 analytic representation of solutions. Through asymptotic analysis, we show that the gradient converges exponentially to zero and that the energy converges 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 results provide a continuous-flow framework in which the Stiefel constraint is recovered asymptotically rather than imposed on the initial data.