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An orthogonality-preserving approach for eigenvalue problems

Tianyang Chu, Xiaoying Dai, Shengyue Wang, Aihui Zhou

TL;DR

The paper tackles the computational burden of large-scale eigenvalue problems with orthogonality constraints by introducing an intrinsic orthogonality-preserving evolution model. The authors establish global well-posedness and energy dissipation, and show that the model converges to the ground-state eigenfunctions, up to orthogonal transformations, with exponential rates under a suitable energy window. They develop an explicit, CFL-free time-stepping scheme that preserves orthogonality and is parallel-friendly, and prove its energy decay and exponential convergence (orbit-wise). Numerical experiments on 2D harmonic oscillator and 3D hydrogen-like Schrödinger equations demonstrate robust, mesh-independent time stepping, energy decay, and orbital-wise exponential convergence, validating the method’s efficiency for computing many eigenpairs in large-scale settings.

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.

An orthogonality-preserving approach for eigenvalue problems

TL;DR

The paper tackles the computational burden of large-scale eigenvalue problems with orthogonality constraints by introducing an intrinsic orthogonality-preserving evolution model. The authors establish global well-posedness and energy dissipation, and show that the model converges to the ground-state eigenfunctions, up to orthogonal transformations, with exponential rates under a suitable energy window. They develop an explicit, CFL-free time-stepping scheme that preserves orthogonality and is parallel-friendly, and prove its energy decay and exponential convergence (orbit-wise). Numerical experiments on 2D harmonic oscillator and 3D hydrogen-like Schrödinger equations demonstrate robust, mesh-independent time stepping, energy decay, and orbital-wise exponential convergence, validating the method’s efficiency for computing many eigenpairs in large-scale settings.

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.

Paper Structure

This paper contains 30 sections, 22 theorems, 188 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Problem settings
  5. An orthogonality-persevering model
  6. The model
  7. Well-posedness
  8. Asymptotic behavior
  9. Time discretization
  10. Numerical scheme
  11. Convergence
  12. Convergence rate
  13. Numerical experiments
  14. Test problem I: Two–dimensional harmonic oscillator equation
  15. Test problem II: Three–dimensional Schrödinger equation for hydrogen atom
  16. ...and 15 more sections

Key Result

Lemma 3.2

For $U \in [L^2(\Omega)]^N$ with $\langle U, U \rangle = I_N$. The matrix is semi-positive definite.

Figures (3)

  • Figure 1: Numerical results for test problem I
  • Figure 2: Numerical results for test problem II
  • Figure 3: Two-dimensional slice heatmap of five eigenvectors: (a) correspond to the first single eigenvalue $\lambda_1$; (b)--(e) correspond to the second eigenvalue $\lambda_2$ with quadruple degenerate.

Theorems & Definitions (51)

  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proof 1
  • Proposition 3.4
  • Proof 2
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 41 more