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Block subspace expansions for eigenvalues and eigenvectors approximation

Francisco Arrieta Zuccalli, Pedro Massey, Demetrio Stojanoff

TL;DR

This work develops a rigorous framework for the optimal expansion of search subspaces to approximate exterior eigenpairs of a large matrix $A$. It proves the existence and construction of stepwise optimal subspace expansions $\mathcal V_t$ via $\mathcal V_{t}=\mathcal V_{t-1}+A(\mathcal W_t)$ and relates these to block Krylov subspaces, establishing convergence bounds in the Hermitian case through polynomial filtering and Chebyshev techniques. The paper also derives computable variants based on Rayleigh-Ritz projections and refined projections, analyzes their convergence, and provides extensive numerical experiments illustrating the trade-offs between optimality, computability, and storage, as well as the impact of spectral gaps and oversampling. Overall, the results offer a principled approach to restarted block-Krylov methods for simultaneous exterior-eigenvalue/eigenvector approximation and guidance for practical projection-based implementations.

Abstract

Let $A\in\mathbb C^{n\times n}$ and let $\mathcal X\subset \mathbb C^n$ be an $A$-invariant subspace with $\dim \mathcal X=d\geq 1$, corresponding to exterior eigenvalues of $A$. Given an initial subspace $\mathcal V\subset \mathbb C^n$ with $\dim \mathcal V=r\geq d$, we search for expansions of $\mathcal V$ of the form $\mathcal V+A(\mathcal W_0)$, where $\mathcal W_0\subset \mathcal V$ is such that $\dim \mathcal W_0\leq d$ and such that the expanded subspace is closer to $\mathcal X$ than the initial $\mathcal V$. We show that there exist (theoretical) optimal choices of such $\mathcal W_0$, in the sense that $θ_i(\mathcal X,\mathcal V+A(\mathcal W_0))\leq θ_i(\mathcal V+A(\mathcal W))$ for every $\mathcal W\subset \mathcal V$ with $\dim \mathcal W\leq d$, where $θ_i(\mathcal X,\mathcal T)$ denotes the $i$-th principal angle between $\mathcal X$ and $\mathcal T$, for $1\leq i\leq d\leq \dim \mathcal T$. We relate these optimal expansions to block Krylov subspaces generated by $A$ and $\mathcal V$. We also show that the corresponding iterative sequence of subspaces constructed in this way approximate $\mathcal X$ arbitrarily well, when $A$ is Hermitian and $\mathcal X$ is simple. We further introduce computable versions of this construction and compute several numerical examples that show the performance of the computable algorithms and test our convergence analysis.

Block subspace expansions for eigenvalues and eigenvectors approximation

TL;DR

This work develops a rigorous framework for the optimal expansion of search subspaces to approximate exterior eigenpairs of a large matrix . It proves the existence and construction of stepwise optimal subspace expansions via and relates these to block Krylov subspaces, establishing convergence bounds in the Hermitian case through polynomial filtering and Chebyshev techniques. The paper also derives computable variants based on Rayleigh-Ritz projections and refined projections, analyzes their convergence, and provides extensive numerical experiments illustrating the trade-offs between optimality, computability, and storage, as well as the impact of spectral gaps and oversampling. Overall, the results offer a principled approach to restarted block-Krylov methods for simultaneous exterior-eigenvalue/eigenvector approximation and guidance for practical projection-based implementations.

Abstract

Let and let be an -invariant subspace with , corresponding to exterior eigenvalues of . Given an initial subspace with , we search for expansions of of the form , where is such that and such that the expanded subspace is closer to than the initial . We show that there exist (theoretical) optimal choices of such , in the sense that for every with , where denotes the -th principal angle between and , for . We relate these optimal expansions to block Krylov subspaces generated by and . We also show that the corresponding iterative sequence of subspaces constructed in this way approximate arbitrarily well, when is Hermitian and is simple. We further introduce computable versions of this construction and compute several numerical examples that show the performance of the computable algorithms and test our convergence analysis.

Paper Structure

This paper contains 13 sections, 15 theorems, 86 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Optimal (block) subspace expansion problem
  5. Analysis of Algorithm \ref{['algo1']}: exterior spectra in the Hermitian case
  6. Computable subspace expansions
  7. Proofs of the main results
  8. Numerical examples
  9. Appendix: some technical results
  10. Principal angles between subspaces
  11. Principal angles and their tangents
  12. Analysis of iterative algorithms
  13. Upper bounds for Chebyshev polynomials

Key Result

Theorem 3.2

Consider Notation notac1. Denote by ${\cal S} := {\cal V} + A({\cal V})$. Then, If we let ${\cal N} := (1-P_{\cal V})(P_{\cal S}(\mathcal{X})\,)$ and $\mathcal{C}_{\rm op} = R^\dagger (\, {\cal N}\,) \subseteq \ker R^\perp \subseteq\mathbb{C}^r \,$ then

Figures (4)

  • Figure 1: Plots of the different (baseline) diagonals considered for the test matrices.
  • Figure 2: Linear decay models
  • Figure 3: Eliptical decay models
  • Figure 4: Polynomial (baseline) decay model with moderate oversampling and various dimensions.

Theorems & Definitions (30)

  • Theorem 3.2
  • Remark 3.3
  • Corollary 3.4
  • Remark 3.5: Algorithm \ref{['algo1']} vs. block Krylov subspace method
  • Theorem 3.8
  • Theorem 3.9
  • Remark 3.10
  • Definition 3.11: Jia22
  • Remark 3.12: Comparison of Algorithm \ref{['algo2']} with previous numerical expansions
  • Proposition 4.1
  • ...and 20 more