A gradient flow-based robust algorithm for solving a sequence of smallest eigenvalues with applications

Abstract

Eigenvalues are core quantities to connect mathematical operators with practical physical and engineering problems. The computation of eigenvalues (and eigen-vectors) is a fundamental numerical procedure with broad and critical applications across engineering, science, and interdisciplinary fields-especially in problems involving dynamic systems, stability analysis, pattern recognition, and multi-physics coupling. In this paper, we first review a few important application examples of eigenvalues in finite element error estimation, energy engineering, quantum mechanics, and chemical adsorption. Existing eigenvalue solving methods have drawbacks such as sensitivity to initial parameters and for iterative methods, lack of robustness with respect to iterations, for eample, difficulty in balancing the preservation of length and unconditional energy stability at the iteration.  In this paper, we present an efficient and robust algorithm for solving the first m eigenvalues problem.  It can be extended to both linear and nonlinear problems. This algorithm has several unique advantages. It converges to the first eigenpair or the first m eigenpairs from arbitrary initial guesses.  Moreover, its strict length preservation and unconditional energy stability ensure high robustness throughout the computation and maintain stable convergence even when large time steps are employed. We begin by deriving the desired gradient flow equation through the introduction of an extended gradient flow and prove that it possesses the properties of length preservation and energy dissipation, implying that the solutions of our gradient flow equation lies  on the Stiefel manifold. Then it is proved that the ordinary differential equation form of this equation has the desired property of existence and uniqueness of a solution. We present an effective algorithm and demonstrate that both its ordinary differential equation and discrete scheme retain length preservation and energy dissipation. Finally, the effectiveness of the algorithm is validated through numerical experiments.

Document Type: Original article

Cited as: Peng, N., Sun, S. A gradient flow-based robust algorithm for solving a sequence of smallest eigenvalues with applications. Computational Energy Science, 2025, 2(2): 24-37. https://doi.org/10.46690/compes.2025.02.03

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