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Lanczos with compression for symmetric eigenvalue problems

Angelo A. Casulli, Daniel Kressner, Nian Shao

TL;DR

This work proposes a novel strategy, called Lanczos with compression, which compresses the Krylov subspace using rational approximation and sacrifices the structure of the associated Krylov decomposition, but remains compatible with subsequent Lanczos steps.

Abstract

The Lanczos method with implicit restarting is one of the most popular methods for finding a few exterior eigenpairs of a large symmetric matrix $A$. Usually based on polynomial filtering, restarting is crucial to limit memory and the cost of orthogonalization. In this work, we propose a novel strategy for the same purpose, called Lanczos with compression. Unlike polynomial filtering, our approach compresses the Krylov subspace using rational approximation and, in doing so, it sacrifices the structure of the associated Krylov decomposition. Nevertheless, it remains compatible with subsequent Lanczos steps and the overall algorithm is still solely based on matrix-vector products with $A$. On the theoretical side, we show that compression introduces only a small error compared to standard (unrestarted) Lanczos and therefore has only a negligible impact on convergence. Comparable guarantees are not available for commonly used implicit restarting strategies, including the Krylov--Schur method. On the practical side, our numerical experiments demonstrate that compression often outperforms the Krylov--Schur method in terms of matrix-vector products.

Lanczos with compression for symmetric eigenvalue problems

TL;DR

This work proposes a novel strategy, called Lanczos with compression, which compresses the Krylov subspace using rational approximation and sacrifices the structure of the associated Krylov decomposition, but remains compatible with subsequent Lanczos steps.

Abstract

The Lanczos method with implicit restarting is one of the most popular methods for finding a few exterior eigenpairs of a large symmetric matrix . Usually based on polynomial filtering, restarting is crucial to limit memory and the cost of orthogonalization. In this work, we propose a novel strategy for the same purpose, called Lanczos with compression. Unlike polynomial filtering, our approach compresses the Krylov subspace using rational approximation and, in doing so, it sacrifices the structure of the associated Krylov decomposition. Nevertheless, it remains compatible with subsequent Lanczos steps and the overall algorithm is still solely based on matrix-vector products with . On the theoretical side, we show that compression introduces only a small error compared to standard (unrestarted) Lanczos and therefore has only a negligible impact on convergence. Comparable guarantees are not available for commonly used implicit restarting strategies, including the Krylov--Schur method. On the practical side, our numerical experiments demonstrate that compression often outperforms the Krylov--Schur method in terms of matrix-vector products.
Paper Structure (33 sections, 7 theorems, 96 equations, 7 figures, 3 tables)

This paper contains 33 sections, 7 theorems, 96 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation.
  3. Algorithm description
  4. Lanczos process and Krylov--Schur restarting
  5. Compression mechanism
  6. Construction of compression matrix $V_{\ell}$
  7. Idealized compression
  8. Injection of rational function
  9. Choice of rational function $r$
  10. Adaptive strategy to avoid tiny gaps
  11. Analysis of compression error
  12. Reorthogonalization with fill-in
  13. Reorthogonalization of Lanczos expansion
  14. Compression step
  15. Backward stability
  16. ...and 18 more sections

Key Result

Theorem 2.1

\newlabelthmLan0 Consider the Krylov-like decomposition eq:LCD for a symmetric matrix $A\in\mathbb{R}^{n\times n}$. Given an orthonormal matrix $V_{\ell} \in \mathbb{R}^{m \times \ell}$, with $\ell< m$, set $\widetilde{Q}_{\ell} := Q_{m}V_{\ell} \in \mathbb{R}^{n\times \ell}$. Then where

Figures (7)

  • Figure 1: Illustration of the parameter choices \ref{['eq:stepfunset', 'eq:paraRA']} for rational approximation.
  • Figure 1: Convergence history of Laplacian eigenvalue problems. The X-axis represents the number of matvecs, and the Y-axis represents the relative error \ref{['eq:errorritz']} of the Ritz values for $k=1$ (left plot) and $k = 4$ (right plot). The maximum dimension of the Krylov subspace is $m = 60$. KS-$\ell$ denotes KS with $\ell$ Ritz vectors retained after restarting.
  • Figure 2: Improvement of LC over KS, as defined in \ref{['defimprovement']} as the size $n$ of discretized Laplacian eigenvalue problem increases. The maximum dimension of the Krylov subspace is set to $60$.
  • Figure 3: Convergence history of Laplacian eigenvalue problems. The X-axis represents the number of matvecs, while the Y-axis shows, for (a), the error of the Ritz values and, for (b), the norm of the residuals, for computing $k = 1$ eigenvalue (left) or $k = 4$ eigenvalues (right). The maximum dimension of the Krylov subspace for LC is set to $60$. The Lanczos method is implemented without restarting or reorthogonalization. The residual is computed after each matvec of LC.
  • Figure 4: Convergence histories of LC with different compression errors, when setting the maximum dimension of the Krylov subspace to $60$. The X-axis represents the number of matvecs.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Theorem 2.1
  • Proof 1
  • Proposition 2.2
  • Proof 2
  • Remark 2.3
  • Lemma 2.4
  • Proof 3
  • Remark 2.5
  • Theorem 2.6
  • Proof 4
  • ...and 7 more