# [Cancelled] Heike Fassbender - A new framework for solving Lyapunov (and other matrix) equations

Dates: | 27 March 2020 |
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Times: | 14:00 - 15:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Speaker: | Heike Fassbender |

Prof. Heike Fassbender from University of Oxford will be speaking at the seminar. Abstract: We will consider model order reduction for stable linear time-invariant (LTI) systems \y=Cx\ with real, large and sparse system matrices. In particular, $A$ is a square $n \times n$ matrix, $B$ is rectangular $n \times m,$ and $C$ is $p \times n.$ Among the many existing model order reduction methods our focus will be on (approximate) balanced truncation. The method makes use of the two Lyapunov equations \A\mathfrak{P}+\mathfrak{P}A^T=-BB^T,\ and \\mathfrak{Q}+\mathfrak{Q}A=-C^TC.\ The solutions $\mathfrak{P}$ and $\mathfrak{Q}$ of these equations are called the controllability and observability Gramians, respectively. The balanced truncation method transforms the LTI system into a balanced form whose controllability and observability Gramians become diagonal and equal, together with a truncation of those states that are both difficult to reach and to observe. One way to solve these large-scale Lyapunov equations is via the Cholesky factor–alternating direction implicit (CF–ADI) method which provides a low rank approximation to the exact solution matrix $\mathfrak{P}$, $\mathfrak{Q}$ resp.. After reviewing existing solution techniques, in particular the CF-ADI method, we will present and analyze a system of ODEs, whose solution for $t \rightarrow \infty$ is the Gramian $\mathfrak{P}.$ We will observe that the solution evolves on a manifold and will characterize numerical methods whose approximate low-rank solution evolves on this manifold as well. This will allow us to give a new interpretation of the ADI method.

### Speaker

Heike Fassbender

Role: Professor of Mathematics

Organisation: AG Numerik Technische Universität Braunschweig