Lanczos with compression for symmetric matrix Lyapunov equations

TL;DR

This paper introduces a compression strategy for the Lanczos method to efficiently solve large-scale symmetric Lyapunov equations, reducing memory usage while maintaining convergence.

Contribution

It proposes a novel Krylov subspace compression technique that enhances the Lanczos method's efficiency for large-scale problems with theoretical and numerical validation.

Findings

Significant memory reduction achieved in Krylov subspace storage.

Maintained convergence rates despite compression.

Validated effectiveness through numerical experiments.

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.