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Stabilizing the Rayleigh--Ritz procedure by randomization

Nian Shao

Abstract

Extracting approximate eigenpairs from a prescribed subspace is of fundamental importance in eigenvalue computation. While projecting the target eigenvector onto the subspace yields satisfactory accuracy, extracting an approximate eigenpair that attains a comparable convergence rate has remained a long-standing open problem. Although the standard Rayleigh--Ritz procedure is widely used for this purpose, it may suffer from deteriorated convergence of Ritz values and may even fail to produce convergent Ritz vectors. In this paper, we address this long-standing open problem by introducing a randomized Rayleigh--Ritz procedure whose output converges at a rate similar to the ideal projection. Our analysis requires only the simplicity of the target eigenvalue and extends naturally to nonlinear eigenvalue problems.

Stabilizing the Rayleigh--Ritz procedure by randomization

Abstract

Extracting approximate eigenpairs from a prescribed subspace is of fundamental importance in eigenvalue computation. While projecting the target eigenvector onto the subspace yields satisfactory accuracy, extracting an approximate eigenpair that attains a comparable convergence rate has remained a long-standing open problem. Although the standard Rayleigh--Ritz procedure is widely used for this purpose, it may suffer from deteriorated convergence of Ritz values and may even fail to produce convergent Ritz vectors. In this paper, we address this long-standing open problem by introducing a randomized Rayleigh--Ritz procedure whose output converges at a rate similar to the ideal projection. Our analysis requires only the simplicity of the target eigenvalue and extends naturally to nonlinear eigenvalue problems.

Paper Structure

This paper contains 18 sections, 10 theorems, 88 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Stability issues of standard Rayleigh--Ritz procedure
  4. RRR procedure for nonlinear eigenvalue problems
  5. Convergence analysis
  6. Exact recovery
  7. Probabilistic analysis on eigenvalue and eigenvector condition numbers
  8. Main result: convergence of randomized Ritz pairs
  9. Refining randomized Ritz values
  10. Refinement via Rayleigh functional.
  11. Refinement via stationary points.
  12. Numerical experiments
  13. Neutral modes in Hamiltonian systems
  14. The butterfly nonlinear eigenvalue problem
  15. Empirical probability and oversampling
  16. ...and 3 more sections

Key Result

Theorem 1

Given a matrix-valued function $B(\cdot)\colon \mathcal{D}\mapsto\mathbb{C}^{m\times m}$ satisfying asp:A. Let $\lambda$ be a simple eigenvalue of $B(\cdot)$ associated with left and right eigenvectors $x$ and $y$. Let $Z_{\perp}$ and $Y_{\perp}$ be orthonormal bases of $\mathsf{span}\{B^{\prime}(\l When $\epsilon$ is small enough, there exists an eigenpair $(\lambda +\Delta \lambda,y+\Delta y)$ o

Figures (3)

  • Figure 1: Convergence of approximate neutral modes extracted from $\mathcal{W}_k$. The X-axis shows the dimension $k$ of $\mathcal{W}_{k}$, while the Y-axis reports the accuracy of approximate eigenpairs measured by \ref{['exp:err']}. Here, "angle" denotes $\angle(v,\mathcal{W}_k)$; "RR-Vec/Val" denote Ritz vectors/values from the standard Rayleigh--Ritz procedure; "RRR-Vec/Val/reVal" denote randomized Ritz vectors/values and refined randomized Ritz values from the RRR procedure (\ref{['alg:RRR']}). The matrix $G_{21}$ in \ref{['eq:defHam']} is zero in the left panel and a complex Gaussian random matrix in the right panel.
  • Figure 2: Convergence of approximate eigenpairs of the butterfly eigenvalue problem extracted from $\mathcal{W}_{k}$. The X-axis shows the dimension $k$ of $\mathcal{W}_{k}$, while the Y-axis reports the accuracy of approximate eigenpairs measured by \ref{['exp:err']}. Here, "angle" denotes $\angle(v,\mathcal{W}_k)$; "RR-Vec/Val" denote Ritz vectors/values from the standard Rayleigh--Ritz procedure; "RRR-Vec/Val/reVal" denote randomized Ritz vectors/values and refined randomized Ritz values from the RRR procedure (\ref{['alg:RRR']}). The matrix $A(\lambda)$ is Hermitian for the left panel, and non-Hermitian for the right panel.
  • Figure 3: Empirical results from $2^{17}$ samples on accuracy (measured by \ref{['exp:err']}) for the RRR procedure (\ref{['alg:RRR']}). Here, "RRR-Vec/Val/reVal" denote randomized Ritz vectors/values and refined randomized Ritz values, and "os" denotes oversampling. The right panel reports the accuracy versus frequency.

Theorems & Definitions (19)

  • Example 1: Interior eigenvalue of a Hermitian matrix: linear convergence of Ritz values and failure of convergence of Ritz vectors
  • Example 2: Non-Hermitian matrix: $\mathcal{O}(\epsilon^{1/m})$ convergence on Ritz values
  • Example 3: Generalized eigenvalue problem: Ritz values fail to converge
  • Remark 1
  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • ...and 9 more