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%Ulrike","baur@mathematik.tu-chemnitz.de","
\section{Model Reduction for unstable Systems based on Hierarchical Matrix Arithmetic} By {\sl Baur, Ulrike and Benner, Peter}.  

\medskip  \noindent We consider linear time-invariant (LTI) systems of the following form \[ \Sigma: \left\{ \begin{array}{rclll}  \dot{x}(t) &=& Ax(t) + Bu(t), & \quad t > 0, & \quad x(0)=x^0, \\  y(t)       &=& Cx(t) + Du(t), & \quad t \geq 0, \end{array} \right. \] with $A\in \mathbf{R}^{n\times n}$,  $B\in \mathbf{R}^{n\times m}$,  and $C\in\mathbf{R}^{p\times n}$ arising, e.g., from the discretization and linearization of parabolic PDEs.  We will assume that the system $\Sigma$ is large-scale with $n \gg m,\, p$ and that the system is unstable, satisfying \[ \Lambda(A)\cap \mathbf{C}^+ \ne \emptyset,\quad \Lambda(A)\cap \jmath\mathbf{R}=\emptyset. \] We further allow the system matrix $A$ to be dense, provided that a {\em data-sparse} representation exists.  To reduce the dimension of the system $\Sigma$, we apply an approach based on the controllability and observability Gramians of $\Sigma$.  The numerical solution of these Gramians is obtained by solving  two algebraic Bernoulli and two Lyapunov equations. As standard methods for the solution of matrix equations are of limited use for large-scale systems, we investigate approaches based on the {\em matrix sign function} method.  To make this iterative method applicable in the large-scale setting,  we incorporate structural information from the underlying PDE model into the approach.  By using data-sparse matrix approximations,  hierarchical matrix formats, and the corresponding  formatted arithmetic we obtain an efficient solver having linear-polylogarithmic complexity.  Once the Gramians are computed, a reduced-order system can be obtained applying the usual {\em balanced truncation method}.
%model reduction, unstable LTI systems, hierarchical matrices","93B40","65F10","talk is part of ""MS5, Linear Algebra in Model Reduction""","04:33:55","Wed Mar 26 2008","134.109.40.166"
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