Inexact methods for the low rank solution to large scale Lyapunov equations

TL;DR

This paper develops a theoretical framework and practical strategies for inexact linear solves within Krylov and ADI methods to efficiently compute low-rank solutions of large-scale Lyapunov equations.

Contribution

It introduces a relaxation strategy for inexact solves in RKSM and LR-ADI, improving computational efficiency for large-scale problems.

Findings

The relaxation strategy maintains solution accuracy while reducing computational cost.

Numerical examples demonstrate the effectiveness of the proposed inexact solve approach.

Theoretical analysis supports the practical implementation of the relaxation method.

Abstract

The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method the repeated solution to a shifted linear system is necessary. For very large systems this solve is usually implemented using iterative methods, leading to inexact solves within this inner iteration. We derive theory for a relaxation strategy within these inexact solves, both for the RKSM and the LR-ADI method. Practical choices for relaxing the solution tolerance within the inner linear system are then provided. The theory is supported by several numerical examples.