AF: Small: Interpolatory Methods for Dimension Reduction of Parametric and Nonlinear Dynamical Systems

William Marsh Rice University, Houston TX

Investigators

Abstract

Dynamical systems are a principal tool in the modeling and control of physical phenomena as diverse as signal propagation in the neural/nervous system, circuit simulation, weather forecasting, and fluid dynamics. Direct numerical simulation has been one of very few available means for studying the rich complexity of these phenomena, and in many areas of engineering numerical simulation has become essential to the design process. However, the ever increasing demand for improved model fidelity leads inevitably to dynamical systems of extremely large scale and complexity. Simulations based on such systems often impose unmanageable burdens on both human and computational resources, and thus provide the principal motivation for model reduction - creating smaller, cheaper models that closely mimic the behaviors of the original system. Model reduction can thus result in tractable low dimensional systems that are suitable for analysis, simulation, optimization, and computer-aided system design. The primary theme of this research is an empirical data approach combined with interpolation to overcome limitations of standard projection methods for linear problems. The empirical data may be provided by physical experimentation or by direct numerical simulation. Interpolation conditions enter in several ways to greatly decrease the computational complexity of the reduced models. A three orders of magnitude reduction in computation time can be achieved while retaining excellent accuracy. The proposed research will strive to put these techniques on firm mathematical foundations in order to assure accuracy and also to greatly extend the areas of application in order to establish broad applicability of these new approaches. The proposed approaches represent a significant departure from existing methodology. Developing the proposed methods to a greater level of maturity and applicability will be a significant advance in model reduction.