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Randomized Implicitly Restarted Arnoldi method for the non-symmetric eigenvalue problem

Jean-Guillaume de Damas, Laura Grigori

Abstract

In this paper, we introduce a randomized algorithm for solving the non-symmetric eigenvalue problem, referred to as randomized Implicitly Restarted Arnoldi (rIRA). This method relies on using a sketch-orthogonal basis during the Arnoldi process while maintaining the Arnoldi relation and exploiting a restarting scheme to focus on a specific part of the spectrum. We analyze this method and show that it retains useful properties of the Implicitly Restarted Arnoldi (IRA) method, such as restarting without adding errors to the Ritz pairs and implicitly applying polynomial filtering. Experiments are presented to validate the numerical efficiency of the proposed randomized eigenvalue solver.

Randomized Implicitly Restarted Arnoldi method for the non-symmetric eigenvalue problem

Abstract

In this paper, we introduce a randomized algorithm for solving the non-symmetric eigenvalue problem, referred to as randomized Implicitly Restarted Arnoldi (rIRA). This method relies on using a sketch-orthogonal basis during the Arnoldi process while maintaining the Arnoldi relation and exploiting a restarting scheme to focus on a specific part of the spectrum. We analyze this method and show that it retains useful properties of the Implicitly Restarted Arnoldi (IRA) method, such as restarting without adding errors to the Ritz pairs and implicitly applying polynomial filtering. Experiments are presented to validate the numerical efficiency of the proposed randomized eigenvalue solver.
Paper Structure (21 sections, 14 theorems, 129 equations, 5 figures, 3 tables, 4 algorithms)

This paper contains 21 sections, 14 theorems, 129 equations, 5 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. The Rayleigh-Ritz procedure and the Arnoldi algorithm
  5. The implicitly restarted Arnoldi method
  6. Sketching
  7. Randomized orthogonalization processes and their usage in Arnoldi
  8. Computing a sketch-orthonormal basis
  9. The randomized Rayleigh-Ritz procedure
  10. Randomized Implicitly Restarted Arnoldi
  11. Main algorithm
  12. Monitoring the residual error
  13. Implicit sketching and sketched vectors storage
  14. Analysis of randomized Arnoldi and randomized Implicitly Restarted Arnoldi
  15. Analysis of randomized Arnoldi
  16. ...and 6 more sections

Key Result

Proposition 2.1

If $A \in \mathbb{R}^{n \times n}$ then there exists an orthonormal $Q \in \mathbb{R}^{n \times n}$ such that where $R \in \mathbb{R}^{n \times n}$ is block upper triangular and has the same eigenvalues as $A$. Its diagonal blocks are of size $1 \times 1$ or $2 \times 2$, the latter accounting for complex conjugate pairs of eigenvalues. The columns of $Q$ are called Schur vectors. They can be cho

Figures (5)

  • Figure 1: Sketched orthonormalization of numerically singular $W \in \mathbb{R}^{10^5 \times 300}$
  • Figure 2: rIRA to compute $k = 10$ Ritz pairs for the non-symmetric $800\times800$ toy matrix $A$ with spectrum $\Lambda_{A} = \{1,2,3,\dots,800\}$ and for two parts of the spectrum, namely largest and smallest modulus. \ref{['fig:diagLM']} and \ref{['fig:diagSM']} : residual norms $\norm*{A \tilde{u}_i - \tilde{\lambda}_i \tilde{u}_i} / \norm*{ \tilde{u}_i}$ along the iterations, \ref{['fig:eivaldiagLM']} and \ref{['fig:eivaldiagSM']} : eigenvalues $\tilde{\lambda}_i$ along the iterations.
  • Figure 3: Stability of orthogonalization processes and resulting eigenvalues for RGS and rCGS2 with $A$ = poli4, $k + p$ = 200 and $k = 100$
  • Figure 4: $A$ = Hamrle3 of size $n \approx 1.4 \times 10^6$, $k = 200$ LM eigenpairs computed in a Krylov dimension $k + p = 500$
  • Figure 5: $A$ = vas_stokes_1M of size $n \approx 1.1 \times 10^6$, $k = 50$ SM eigenpairs computed in a Krylov dimension $k + p = 100$

Theorems & Definitions (31)

  • Definition 2.1: Rayleigh-Ritz approximation for eigenpairs using Arnoldi iteration
  • Proposition 2.1: real Schur decomposition
  • Theorem 2.1: from Theorem 2.9 of Sorensen1992ImplicitApplicationPolynomial
  • Definition 2.2: $\varepsilon$-embedding
  • Definition 2.3: Oblivious subspace embedding (OSE)
  • Corollary 2.1.1
  • Definition 3.1: randomized Arnoldi factorization
  • Definition 3.2: randomized Rayleigh-Ritz using randomized Arnoldi
  • Definition 4.1
  • Theorem 4.1
  • ...and 21 more