Model Reduction for Structured Dynamical Systems

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This project is concerned with the investigation of five specific topics in model reduction. (a) Decay rates of certain system related singular values (Hankel singular values and others related to approximation error bounds); (b) Model reduction for passive systems; (c) Convergence of Krylov-like projection algorithms for model reduction and the establishment of error bounds for such methods; (d) Reduction methods for periodically time-varying systems, and (e) Structure preserving reduction methods for second order dynamical systems.Model reduction seeks to replace a large-scale system of differential or difference equations by a system of substantially lower dimension, that ideally, has the same response characteristics as the original system, yet requires far less computational resources for realization. Such large-scale systems arise in circuit simulation; they also arise through spatial discretization of certain time dependent PDE control systems and in many other applications. For example, an important step in chip manufacturing is the physical verification step, where a detailed simulation, modeling all constituent components of the chip must be carried out to check its behavior. Full simulation is out of the question due to computational complexity. Simulation based upon a reduced model is required to complete the computation in a reasonable period of time. However, it is essential that accuracy of the results is sufficient and that salient physical properties of the chip are faithfully preserved with the reduced model. This research is focused on the development, analysis, and implementation of reduction methods for very large problems. Where needed, the work will involve extending the underlying theory of dimension reduction, particularly for control problems. The primary goal is to provide reliable and efficient dimension reduction methods that preserve structure and system properties with rigorously established bounds on approximation error.