Peter Benner - Model Reduction of Complex Dynamical Systems

This volume contains papers related to presentations given there.

In On Bilinear Time Domain Identification and Reduction in the Loewner Framework, D. S. Karachalios, I. V. Gosea, and A. C. Antoulas propose a two-step procedure that learns a reduced bilinear control system based on time-domain measurements. In a first step, a linear surrogate model is fitted which subsequently is extended by a suitably fitted bilinear operator. The approach combines the Loewner framework with Volterra series representations.

Large-scale linear control systems are routinely tackled with the model reduction method of balanced truncation. Balanced truncation requires solving a pair of Lyapunov equations for two Gramians associated with the system. The work Balanced Truncation for Parametric Linear Systems Using Interpolation of Gramians: a Comparison of Algebraic and Geometric Approaches by N. T. Son, P.-Y. Gousenbourger, E. Massart, and T. Stykel addresses such linear control systems in the presence of additional parameter dependencies. Their approach to parametric MOR is to approximate the Gramians of the Lyapunov equations at a given parameter via interpolation. Two methods are proposed: direct matrix interpolation (called the algebraic approach) and interpolation of the Gramians on the matrix manifold of positive semidefinite matrices of fixed rank (called the geometric approach). Both methods are juxtaposed and assessed by means of numerical examples.

Dynamic mode decomposition (DMD) is a data-driven method for learning the dynamics of complex nonlinear systems. This information, in turn, can be exploited to construct reduced-order models. In the contribution