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Sketched and Truncated Polynomial Krylov Subspace Methods: Matrix Sylvester Equations

Davide Palitta, Marcel Schweitzer, Valeria Simoncini

TL;DR

The paper tackles large-scale Sylvester equations $\mathbf{A}\mathbf{X}+\mathbf{X}\mathbf{B}=\mathbf{C}_1\mathbf{C}_2^{\top}$ by introducing a sketched-and-truncated polynomial Krylov approach that leverages oblivious subspace embeddings to reduce orthogonalization cost and memory. Central ideas include WS-Arnoldi whitening, a reduced Galerkin projection for $\mathbf{Y}_d$, a two-pass reconstruction to recover the full low-rank solution, and a theoretical framework based on effective fields of values to ensure well-posedness and convergence. Theoretical results cover field-of-values distortions, Lyapunov-convergence bounds under sketching, and bounds on the distance to full Arnoldi approximations. Numerical experiments demonstrate substantial memory and time savings, competitive performance against RKSM, and robustness to non-orthogonal bases, confirming the practical viability of sketched-and-truncated Krylov projections for large matrix equations.

Abstract

Thanks to its great potential in reducing both computational cost and memory requirements, combining sketching and Krylov subspace techniques has attracted a lot of attention in the recent literature on projection methods for linear systems, matrix function approximations, and eigenvalue problems. Applying this appealing strategy in the context of linear matrix equations turns out to be far more involved than a straightforward generalization. These difficulties include analyzing well-posedness of the projected problem and deriving possible error estimates depending on the sketching properties. Further computational complications include the lack of a natural residual norm estimate and of an explicit basis for the generated subspace. In this paper we propose a new sketched-and-truncated polynomial Krylov subspace method for Sylvester equations that aims to address all these issues. The potential of our novel approach, in terms of both computational time and storage demand, is illustrated with numerical experiments. Comparisons with a state-of-the-art projection scheme based on rational Krylov subspaces are also included.

Sketched and Truncated Polynomial Krylov Subspace Methods: Matrix Sylvester Equations

TL;DR

The paper tackles large-scale Sylvester equations by introducing a sketched-and-truncated polynomial Krylov approach that leverages oblivious subspace embeddings to reduce orthogonalization cost and memory. Central ideas include WS-Arnoldi whitening, a reduced Galerkin projection for , a two-pass reconstruction to recover the full low-rank solution, and a theoretical framework based on effective fields of values to ensure well-posedness and convergence. Theoretical results cover field-of-values distortions, Lyapunov-convergence bounds under sketching, and bounds on the distance to full Arnoldi approximations. Numerical experiments demonstrate substantial memory and time savings, competitive performance against RKSM, and robustness to non-orthogonal bases, confirming the practical viability of sketched-and-truncated Krylov projections for large matrix equations.

Abstract

Thanks to its great potential in reducing both computational cost and memory requirements, combining sketching and Krylov subspace techniques has attracted a lot of attention in the recent literature on projection methods for linear systems, matrix function approximations, and eigenvalue problems. Applying this appealing strategy in the context of linear matrix equations turns out to be far more involved than a straightforward generalization. These difficulties include analyzing well-posedness of the projected problem and deriving possible error estimates depending on the sketching properties. Further computational complications include the lack of a natural residual norm estimate and of an explicit basis for the generated subspace. In this paper we propose a new sketched-and-truncated polynomial Krylov subspace method for Sylvester equations that aims to address all these issues. The potential of our novel approach, in terms of both computational time and storage demand, is illustrated with numerical experiments. Comparisons with a state-of-the-art projection scheme based on rational Krylov subspaces are also included.
Paper Structure (15 sections, 10 theorems, 81 equations, 4 figures, 5 tables, 2 algorithms)

This paper contains 15 sections, 10 theorems, 81 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background material
  3. The block Arnoldi relation
  4. Oblivious subspace embeddings
  5. The sketched Arnoldi relation
  6. Sketched and truncated Krylov methods for matrix equations
  7. Distance from the Arnoldi approximation
  8. Theoretical investigation
  9. On the field of values of the projected problem
  10. Convergence analysis for the Lyapunov setting
  11. The algorithms
  12. Numerical experiments
  13. Conclusions
  14. Algorithmic description of truncated Arnoldi
  15. Tensorized subspace embeddings

Key Result

Proposition 2.1

[Proposition 3.1 in PalittaSchweitzerSimoncini2023, adapted to the block setting] Let $\boldsymbol{S}_{\boldsymbol{U}}\boldsymbol{U}_{d+1}=\boldsymbol{Q}_{\boldsymbol{U},d+1} \boldsymbol{T}_{\boldsymbol{U},d+1}$ be a reduced QR decomposition with Then, for the sketched method, the following Arnoldi-like formula holds: with $R_{H}=\boldsymbol{T}_{\boldsymbol{U},d}^{-1}T_{H}\bm{h}_{d+1,d}$ and $Q_

Figures (4)

  • Figure 1: \ref{['ex:fov1']}. Fields of values $\partial W(\boldsymbol{V}^\top\!\!\boldsymbol{A}\boldsymbol{V})$ (solid blue line) and $\partial W(\boldsymbol{V}^\top\!\boldsymbol{S}^\top\!\boldsymbol{S}\boldsymbol{A}\boldsymbol{V})$ (dashed black line), together with the corresponding eigenvalues.
  • Figure 2: \ref{['first_example']}. Left: $\partial W(\boldsymbol{\widehat{H}}_d)$ (blue solid line), spectra of $\boldsymbol{\widehat{H}}_d$ (black circles), and $\boldsymbol{\widehat{H}}_d+\widehat{h}e_d^\top$ (red crosses). Center: $\partial W(\boldsymbol{\widehat{H}}_d+\widehat{h}e_d^\top)$ (blue solid line) and spectrum of $\boldsymbol{\widehat{H}}_d+\widehat{h}e_d^\top$ (red crosses). Right: $\partial W(\mathcal{Q}_1^*(\boldsymbol{\widehat{H}}_d+\widehat{h}e_d^\top)\mathcal{Q}_1)$ (blue solid line) and spectrum of $\mathcal{Q}_1^*(\boldsymbol{\widehat{H}}_d+\widehat{h}e_d^\top)\mathcal{Q}_1$ (red crosses).
  • Figure 3: \ref{['Example 2']}. Field of values of the coefficient matrix (solid line), corresponding eigenvalues (approximately vertically aligned crosses) and field of values of the reduced matrices, for $d=10, 20, 30, 40$ and $r=1$ (expanding dashed lines as $d$ grows). Left: $\boldsymbol{A}$ and $\widehat{\boldsymbol{H}}_d+\widehat{H}E_d^{\top}$. Right: $\boldsymbol{B}$ and $\widehat{\boldsymbol{G}}_d+\widehat{G} E_d^{\top}$.
  • Figure 4: \ref{['Example 2']}. Scaled condition number of the bases $\mathbf{U}_d$ (red line with circles) and $\mathbf{V}_d$ (blue line with stars), along with the relative Frobenius norm of the residual matrix $\mathbf{R}_d$ (black solid line).

Theorems & Definitions (16)

  • Proposition 2.1
  • Proposition 3.1
  • Proposition 4.1
  • Example 4.2
  • Lemma 4.3
  • Proposition 4.4
  • Example 4.5
  • Theorem 4.6
  • Corollary 4.7
  • Example 6.1
  • ...and 6 more