On an integrated Krylov-ADI solver for large-scale Lyapunov equations

TL;DR

This paper introduces an integrated Krylov-ADI solver that efficiently solves large-scale Lyapunov equations by constructing a unified approximation space, significantly improving computational performance over traditional methods.

Contribution

The authors propose a novel merged iterative approach combining Krylov subspace methods with the ADI algorithm for large Lyapunov equations, enabling efficient solution of shifted linear systems.

Findings

Reduced computational time compared to sparse direct solvers

Effective integration of shift computation algorithms

Demonstrated efficiency on large-scale problems

Abstract

One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efficient implementation of the low-rank ADI method, for an important class of equations. Many state-of-the-art algorithms for the shift computation can be easily incorporated into our new scheme, as well. Several numerical results illustrate the potential of our novel procedure when compared to an implementation of the low-rank ADI method based on sparse direct solvers for the shifted linear systems.