A tangential low-rank ADI method for solving indefinite Lyapunov equations

TL;DR

This paper introduces a new tangential low-rank ADI method for efficiently solving large-scale indefinite Lyapunov equations, especially when the constant term has high rank, with adaptive parameter selection.

Contribution

It proposes a novel tangential reformulation of the ADI iteration that improves efficiency for high-rank indefinite Lyapunov equations and includes adaptive parameter selection methods.

Findings

The method efficiently computes low-rank solutions for high-rank indefinite Lyapunov equations.

Numerical examples demonstrate the effectiveness and robustness of the proposed approach.

Adaptive parameter selection enhances the method's applicability across various problem settings.

Abstract

Continuous-time algebraic Lyapunov equations have become an essential tool in various applications. In the case of large-scale sparse coefficient matrices and indefinite constant terms, indefinite low-rank factorizations have successfully been used to allow methods like the alternating direction implicit (ADI) iteration to efficiently compute accurate approximations to the solution of the Lyapunov equation. However, classical block-type approaches quickly increase in computational costs when the rank of the constant term grows. In this paper, we propose a novel tangential reformulation of the ADI iteration that allows for the efficient construction of low-rank approximations to the solution of Lyapunov equations with indefinite right-hand sides even in the case of constant terms with higher ranks. We provide adaptive methods for the selection of the corresponding ADI parameters, namely shifts and tangential directions, which allow for the automatic application of the method to any relevant problem setting. The effectiveness of the developed algorithms is illustrated by several numerical examples.