Advanced Projection Techniques for Dimension Reduction of Large Scale Dynamical Systems

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Model reduction seeks to replace a large-scale system of differential equations by a system of substantially lower dimension that has response characteristics similar to the original system while requiring far less computational resources. Such large-scale systems often arise through spatial discretization of time dependent PDE systems. For example, in chip manufacturing the physical verification step involves detailed simulation of all constituent components of the chip to verify its behavior. Full simulation is intractable due to computational complexity. A reduced model with guaranteed accuracy is required to complete a reliable simulation in a reasonable period of time. Additional applications of broad impact include: weather prediction, air quality management, micro-electro-mechanical systems, and many others.In general terms, 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. The primary goal is to provide reliable and efficient dimension reduction methods that preserve structure and system properties with bounds on the approximation error. More specifically, this project is concerned with the investigation of four topics in model reduction. (a) Selection of interpolation points in rational Krylov methods with new applications in reduction methods which (i) preserve dissipativity, and (ii) are optimal in the H2 norm. (b) Provably convergent parameter free large scale Lyapunov solvers with applications to approximate balanced truncation with rigorous error bounds.(c) Symmetry preserving principal component analysis for dimension reduction based upon a new symmetry preserving singular value decomposition; and (d) domain decomposition techniques for rapid solution and dimension reduction of large scale systems with application to power grids of VLSI chips and complex building models.