Speaker
Description
Continuous-time algebraic Lyapunov equations are linear matrix equations of the form
$$
\begin{equation*}
A X E^{\mathsf{H}} + E X A^{\mathsf{H}} = -W
\end{equation*}
$$
where $A, E \in \mathbb{C}^{n \times n}$ are large-scale sparse coefficient matrices and $W = B R B^{\mathsf{H}}$ represents an indefinite right-hand side defined by the low-rank factor $B \in \mathbb{C}^{n \times m}$ and the indefinite Hermitian matrix $R = R^{\mathsf{H}} \in \mathbb{C}^{m \times m}$. Such matrix equations have become an essential tool in various applications like model reduction, control or data-driven modeling. 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 work, 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.