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.
Data-Driven Dimension Reduction of Linear and Nonlinear Systems: Dynamical systems are a principal tool in the modeling and control of physicalphenomena 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 studyingthe rich complexity of these phenomena, and in many areas of engineering numerical simulation has become essential to the design process. Optimaldesign of such phenomena is virtually impossible without numerical simulation. However, the ever increasing demand for improved model fidelity leads inevitably to dynamical systems of extremely large scale and complexity. At the same time the systems to be modeled arising mostly from the spatial discretization of the underlying PDEs (partial differential equations), have become more heterogeneousand complex, frequently arising in multi-physics environments. The resulting model manipulations and simulations often imposeunmanageable burdens on both human and computational resources, and thusprovide the principal motivation for model reduction - creating smaller,cheaper models that closely mimic the original. The use of such complex models for analysis, simulation, optimization, and computer-aided system design, can be facilitated (or even made possible)by means of model reduction. Thereby the number of equations involved in the numerical simulation is reduced. The resulting system can then be usedin design, optimization, etc., significantly reducing the development time. Thus, model reduction plays an important rolein simulations which take place in many application areas. We have contributed to recent significant advances in model reduction whichhave positioned the area for breakthroughs that would permit applicationto large-scale systems involving nonlinear dynamics, parameterdependent processes, and systems with many external ports.Our major contributions have been the development and analysis of 1) TheDiscrete Empirical Interpolation Method (DEIM) for nonlinear model reduction and 2) The Lowner Framework for data driven model reduction.Our goal with this project has been to apply and develop model reductionalgorithms within the context of applications, such as:signal propagation in electrical circuits and VLSI chips, neural modelingand computational fluid dynamics. We have made significantadvances in each of these areas and are working to bring thistechnology to a level where it will have broad impact on a muchwider class of problems. We have adopted an empirical data approach combined with interpolationto overcome limitations of standard projection methods for linearproblems. The empirical data may be provided by physical experimentationor by direct numerical simulation. Our common theme is to construct reduced order models from this data. Interpolationconditions enter in several ways to greatly decrease the computational complexity of the reduced models. For example, in a miscible two phase flow in porous media problem, we have successfully reduced a 15,000 variable modelto a 40 dimensional model resulting in a reduction on the order of 1000 in computation time while retaining excellent accuracy. During the course of this project wehave began to put these techniques on firm mathematical foundationsin order to assure accuracy. We have also worked to greatly extend the areas of application in order to establish broad applicability of our approaches. Intellectual merit of the research activity: Our approaches represent a significant departure from existing methodology, especially in the linear regime.For nonlinear and even for time dependent problems, verylittle is available. Hardly any of the techniques developedfor linear time invariant systems apply in these more challengingregimes. Our development of the methods for the nonlinearregime has provided significant advances for model reduction. Broader impact resulting from the research activity: Our work has established the great promise that thesenew approaches have to be broadly applicable in real applicationareas. We have demonstrated effectiveness on very difficultproblems. Already we have made contributions to neural modeling,electromagnetic simulation, VLSI design and fluid dynamics. Many applications require multiple simulations at various parametersettings, changing material properties, etc. The proposedwork will have great impact on such situations. Key words: model order reduction, interpolation methods, proper orthogonal decomposition, Loewner matrices, empirical interpolation, parametric systems.
