Preliminary list of abstracts

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On low-rank solutions of large-scale generalized Lyapunov equations arising in parametric model reduction
Keywords: Parametric model order reduction
Mon, 15:45--16:10
  • Benner, Peter (Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Germany)
  • Breiten, Tobias (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany)

Parametric model order reduction (PMOR) has become an important tool in computing surrogate models of parameter-dependent, dynamical systems to be used in design optimization and control. PMOR can be considered as an approximation problem for a special multi-parametric function. For this purpose, we employ a connection between linear parametric and bilinear control systems which allows for several efficient approximation techniques. Here, we will focus on a generalization of the method of balanced truncation for linear systems which leads to large-scale Lyapunov equations of the form $$AX+XA^T+\sum_{j=1}^mN_jXN_j^T+BB^T=0,$$ with $A,N_j \in \mathbb R^{n\times n}$ and $B \in \mathbb R^{n\times m}.$ Imposing some assumptions on $N_j$ and $B,$ we will make use of the corresponding tensorized version $$\left(I\otimes A +A\otimes I + \sum_{j=1}^mN_j\otimes N_j\right)\operatorname{vec}{(X)}=-\operatorname{vec}{(BB^T)}$$ to show the existence of low-rank approximations to the solution matrix $X.$ Finally, this will justify extending low-rank methods like e.g. the ADI iteration or other rational Krylov subspace methods which are known to be effective tools in the linear case. By means of a numerical example, we will evaluate the performance of those methods.