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A quasi-orthogonal iterative method for eigenvalue problems

Shengyue Wang, Aihui Zhou

TL;DR

The work addresses large-scale eigenvalue problems requiring many orthogonal eigenvectors by introducing a predictor-corrector quasi-orthogonal scheme that avoids explicit orthogonalization and maintains quasi-orthogonality under perturbations. The method operates on discretized Stiefel/Grassmann frameworks and proves range invariance, exponential convergence to orthogonality, and energy-decay-driven convergence of the gradient, energy, and iterates. Numerical tests on elliptic and quantum-mechanical operators demonstrate monotone energy decay with exponential rates, robust orthogonality preservation, and component-wise convergence, all while enabling scalable parallel computation. Practically, a feasible implementation uses a two-step update with a low-rank inverse to sidestep implicit solves, broadening applicability to large-scale problems in physics, chemistry, and data-driven contexts.

Abstract

For large-scale eigenvalue problems requiring many mutually orthogonal eigenvectors, traditional numerical methods suffer substantial computational and communication costs with limited parallel scalability, primarily due to explicit orthogonalization. To address these challenges, we propose a quasi-orthogonal iterative method that dispenses with explicit orthogonalization and orthogonal initial data. It inherently preserves quasi-orthogonality (the iterates asymptotically tend to be orthogonal) and enhances robustness against numerical perturbations. Rigorous analysis confirms its energy-decay property and convergence of energy, gradient, and iterate. Numerical experiments validate the theoretical results, demonstrate key advantages of strong robustness and high-precision numerical orthogonality preservation, and thereby position our iterative method as an efficient, stable alternative for large-scale eigenvalue computations.

A quasi-orthogonal iterative method for eigenvalue problems

TL;DR

The work addresses large-scale eigenvalue problems requiring many orthogonal eigenvectors by introducing a predictor-corrector quasi-orthogonal scheme that avoids explicit orthogonalization and maintains quasi-orthogonality under perturbations. The method operates on discretized Stiefel/Grassmann frameworks and proves range invariance, exponential convergence to orthogonality, and energy-decay-driven convergence of the gradient, energy, and iterates. Numerical tests on elliptic and quantum-mechanical operators demonstrate monotone energy decay with exponential rates, robust orthogonality preservation, and component-wise convergence, all while enabling scalable parallel computation. Practically, a feasible implementation uses a two-step update with a low-rank inverse to sidestep implicit solves, broadening applicability to large-scale problems in physics, chemistry, and data-driven contexts.

Abstract

For large-scale eigenvalue problems requiring many mutually orthogonal eigenvectors, traditional numerical methods suffer substantial computational and communication costs with limited parallel scalability, primarily due to explicit orthogonalization. To address these challenges, we propose a quasi-orthogonal iterative method that dispenses with explicit orthogonalization and orthogonal initial data. It inherently preserves quasi-orthogonality (the iterates asymptotically tend to be orthogonal) and enhances robustness against numerical perturbations. Rigorous analysis confirms its energy-decay property and convergence of energy, gradient, and iterate. Numerical experiments validate the theoretical results, demonstrate key advantages of strong robustness and high-precision numerical orthogonality preservation, and thereby position our iterative method as an efficient, stable alternative for large-scale eigenvalue computations.
Paper Structure (13 sections, 14 theorems, 169 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 14 theorems, 169 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Eigenvalue problem
  4. Notation
  5. A quasi-orthogonal algorithm
  6. Iterative scheme
  7. Range invariance
  8. Quasi-orthogonality
  9. Convergence
  10. Numerical experiments
  11. Concluding remarks
  12. Practical iteration
  13. Additional experimental results

Key Result

Lemma 3.1

\newlabelextend gradient: norm persevation0 Given $n \in \mathbb{N}$, if $U_n$ is an iterate generated by Algorithm alg:Discretization scheme, then and the spectrum of $\tilde{U}_{n+\frac{1}{2}}^\top \tilde{U}_{n+\frac{1}{2}}$ satisfies where Here, $\mathcal{I}$ denotes the identity operator on $\left(\mathcal{V}^{N_g}\right)^N$.

Figures (7)

  • Figure 1: Quasi-orthogonality of the scheme (\ref{['Discretization scheme']})
  • Figure 1: Numerical results of Example \ref{['eq:3D Laplace eigenvalue equation']}
  • Figure 1: Numerical results of Example \ref{['eq:3D Laplace eigenvalue equation']} with special initial data
  • Figure 2: Numerical results of Example \ref{['eq:3D harmonic oscillator equation 3D']}
  • Figure 2: Numerical results of Example \ref{['eq:3D harmonic oscillator equation 3D']} with special initial data
  • ...and 2 more figures

Theorems & Definitions (33)

  • Lemma 3.1
  • Lemma 3.2
  • Proof 1
  • Remark 3.3
  • Lemma 3.4
  • Proof 2
  • Theorem 3.5
  • Proof 3
  • Remark 3.7
  • Theorem 3.8
  • ...and 23 more