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Lanczos with compression for symmetric matrix Lyapunov equations

Angelo A. Casulli, Francesco Hrobat, Daniel Kressner

TL;DR

The paper addresses solving large-scale symmetric Lyapunov equations $A X + X A = \boldsymbol{c}\boldsymbol{c}^T$ with SPD $A$ by leveraging a Lanczos-based Krylov framework augmented with rational approximation. It introduces a compression strategy that maintains convergence while dramatically reducing memory usage, employing cycles and implicit updates to form $Q_s U_s$ without storing the full Lanczos basis. The authors provide rigorous error and residual bounds, including finite-precision analyses for both standard and compressed Lanczos, and they validate the method through numerical experiments on a 4D Laplacian and two model-order reduction problems, showing practical memory and time advantages over existing low-memory approaches. This work enhances the scalability of low-rank Lyapunov solvers, with immediate impact on control, MOR, and PDE-based discretizations where matrix-vector access to $A$ is cheap but full decompositions are infeasible.

Abstract

This work considers large-scale Lyapunov matrix equations of the form $AX + XA = \boldsymbol{c}\boldsymbol{c}^T$, where $A$ is a symmetric positive definite matrix and $\boldsymbol{c}$ is a vector. Motivated by the need to solve such equations in a wide range of applications, various numerical methods have been developed to compute low-rank approximations of the solution matrix $X$. In this work, we focus on the Lanczos method, which has the distinct advantage of requiring only matrix-vector products with $A$, making it broadly applicable. However, the Lanczos method may suffer from slow convergence when $A$ is ill-conditioned, leading to excessive memory requirements for storing the Krylov subspace basis generated by the algorithm. To address this issue, we propose a novel compression strategy for the Krylov subspace basis that significantly reduces memory usage without hindering convergence. This is supported by both numerical experiments and a convergence analysis. Our analysis also accounts for the loss of orthogonality due to round-off errors in the Lanczos process.

Lanczos with compression for symmetric matrix Lyapunov equations

TL;DR

The paper addresses solving large-scale symmetric Lyapunov equations with SPD by leveraging a Lanczos-based Krylov framework augmented with rational approximation. It introduces a compression strategy that maintains convergence while dramatically reducing memory usage, employing cycles and implicit updates to form without storing the full Lanczos basis. The authors provide rigorous error and residual bounds, including finite-precision analyses for both standard and compressed Lanczos, and they validate the method through numerical experiments on a 4D Laplacian and two model-order reduction problems, showing practical memory and time advantages over existing low-memory approaches. This work enhances the scalability of low-rank Lyapunov solvers, with immediate impact on control, MOR, and PDE-based discretizations where matrix-vector access to is cheap but full decompositions are infeasible.

Abstract

This work considers large-scale Lyapunov matrix equations of the form , where is a symmetric positive definite matrix and is a vector. Motivated by the need to solve such equations in a wide range of applications, various numerical methods have been developed to compute low-rank approximations of the solution matrix . In this work, we focus on the Lanczos method, which has the distinct advantage of requiring only matrix-vector products with , making it broadly applicable. However, the Lanczos method may suffer from slow convergence when is ill-conditioned, leading to excessive memory requirements for storing the Krylov subspace basis generated by the algorithm. To address this issue, we propose a novel compression strategy for the Krylov subspace basis that significantly reduces memory usage without hindering convergence. This is supported by both numerical experiments and a convergence analysis. Our analysis also accounts for the loss of orthogonality due to round-off errors in the Lanczos process.

Paper Structure

This paper contains 24 sections, 8 theorems, 66 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Lanczos method combined with rational approximation
  3. Lanczos method
  4. Rational Krylov subspaces
  5. Reference Method
  6. Error Bounds
  7. Pole selection
  8. Computation of the residual
  9. Main algorithm
  10. Notation and derivation of algorithm
  11. Partitioning Lanczos basis and tridiagonal matrix into cycles
  12. Recursive computation of rational Krylov subspace bases
  13. Recursive computation of $Q_s U_s$ and residual estimation
  14. Recursive computation of $S_i$
  15. Theoretical results
  16. ...and 9 more sections

Key Result

Lemma 2.2

Consider Algorithm alg: naive applied to a symmetric positive definite matrix $A\in\mathbb{R}^{N\times N}$, with the smallest and largest eigenvalues of $A$ denoted by $\lambda_{\min}$ and $\lambda_{\max}$, respectively. Suppose that none of the poles $\xi_i$ is in $[\lambda_{\min},\lambda_{\max}]$, Then the error between the solution $X_M$ of the projected equation eqn: lyapproj and its approxima

Figures (2)

  • Figure 1: Graphical representation of the orthonormal basis $Q_{i+1}$ computed until cycle $i+1$.
  • Figure 2: Graphical representation of the tridiagonal matrices $T_{i+1}$ generated until cycle $i+1$.

Theorems & Definitions (17)

  • Definition 2.1: Rational Krylov subspace
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 7 more