Model reduction and real-time control find applications in diverse areas. These include simulation and control of large-scale structures, weather prediction, air quality management, molecular dynamics simulations, simulation and control of chemical reactors (e.g. Chemical Vapor Deposition), and simulation and control of micro-electro-mechanical systems (e.g., micromirrors), to name but a few. In today's technological world, physical processes are described primarily using mathematical models that are used to simulate the behavior of the underlying processes. Often, they are also used to modify or control system behavior. In this framework, there is an ever increasing need for improved accuracy, which leads to models of higher complexity. The basic motivation for system approximation is the need, in many instances, for a simplified model of a dynamical system, which captures the main features of the original complex model. This need arises from limited computational, accuracy, or storage capabilities. The simplified model may then be used in place of the original complex model, either for simulation, or control. As sensor networks and embedded processors proliferate our environment, technologies for such approximations and real-time control emerge as the next major technical challenge.
Model reduction seeks to replace a large-scale system of differential or difference equations by a system of substantially lower dimension that has nearly the same response characteristics. Two main themes can be identified among several methodologies: (i) balanced and Hankel norm methods, and (ii) moment matching methods. Balanced and Hankel norm approximation methods are built upon a family of ideas with very close connection to the singular value decomposition. These methods preserve asymptotic stability and allow for global error bounds. However, they do not scale well in terms of computational efficiency and numerical stability when applied to large scale problems as they rely upon dense matrix computations. Moment matching methods are based principally on Padé-like approximations and for large-scale problems have led naturally to the use of Krylov and rational Krylov subspace projection methods. These methods generally enjoy greater efficiency and numerical stability though maintaining asymptotic stability in the reduced order model cannot be guaranteed and therefore may be problematic at times. Moreover, no global error bounds exist.
A strong current trend aims at combining these two approaches by deriving iterative methods that achieve approximate balanced reduction and which are amenable to implementation on large-scale parallel and distributed platforms. The expertise of the different groups collaborating in this proposal are precisely complementary in this respect. Our current projects have been focused on various aspects of the goals of this proposal. We are proposing to extend the methods we have developed for model reduction of large-scale Linear Time Invariant (LTI) systems to a new regime of problems that require adaptive models. In particular, we will consider large-scale structured problems that are either time-varying, or which require adaptive updating of the initial reduced models to obtain better approximation properties. We also outline key applications that will benefit significantly from the proposed research. Ultimately, we intend to utilize these new reduced order modeling techniques to design low-order real-time controllers for large-scale dynamical systems.
With this background, this project aims to address the following specific research , development, and educational goals:
- The development and implementation of a library of parallel numerical linear algebra algorithms and kernels for those sparse matrix computations that arise in model reduction schemes for large-scale dynamical systems.
- The development and implementation of robust parallel algorithms for open problems in model reduction for structured dynamical systems.
- Validation and testing, primarily for dynamical systems that govern the motion of large-scale structures that are subject to wind, earthquakes, or man-made hazards.
- The development of an effective education and outreach program.
Word Document of Mission Statement