Advanced Projection Techniques for Dimension Reduction of Large Scale Dynamical Systems

模型约简寻求用一个具有与原始系统相似的响应特征的低维系统来取代一个大规模的微分方程系统，同时需要更少的计算资源。这种大尺度系统通常是通过时间相关PDE系统的空间离散化而产生的。例如，在芯片制造中，物理验证步骤包括对芯片所有组成部件的详细模拟，以验证其行为。由于计算的复杂性，完全模拟是难以处理的。为了在合理的时间内完成可靠的仿真，需要一个保证精度的简化模型。其他具有广泛影响的应用包括：天气预报、空气质量管理、微机电系统等。总的来说，本研究的重点是针对非常大的问题的约简方法的开发、分析和实施。在需要的地方，这项工作将涉及扩展基本的降维理论。主要目标是提供可靠和有效的降维方法，以保持结构和系统的性质，并对近似误差有界。更具体地说，该项目涉及模型简化中的四个主题的调查。(a)在合理Krylov方法中选择插值点，并在(i)保持耗散率，（ii）在H2范数中最优的约简方法中有新的应用。(b)可证明收敛的无参数大规模Lyapunov解，并应用于具有严格误差界的近似平衡截断。(c)基于一种新的保对称奇异值分解的保对称主成分降维分析；(d)用于大规模系统快速求解和降维的域分解技术，并应用于VLSI芯片的电网和复杂的建筑模型。

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.