Simulation, design, and control of dynamical systems play an important role in numerous scientific and industrial tasks. The need for detailed models leads to large-scale dynamical systems, posing tremendous computational difficulties when employed in numerical simulations. In order to overcome these challenges, we perform model reduction, replacing the large-scale dynamics with high-fidelity reduced representations. There exist a plethora of methods for reduced order modeling of linear systems, including the Iterative Rational Krylov Algorithm (IRKA), Balanced Truncation (BT), and Hankel Norm Approximation. However, these methods generally target stable systems and the approximation is performed over an infinite time horizon. If we are interested in a finite horizon reduced model, we utilize techniques such as Time-limited Balanced Truncation (TLBT) and Proper Orthogonal Decomposition (POD). In this dissertation we establish interpolation-based optimality conditions over a finite horizon and develop an algorithm, Finite Horizon IRKA (FHIRKA), that produces a locally optimal reduced model on a specified time-interval. Nonetheless, the quantities being interpolated and the interpolant are not the same as in the infinite horizon case. Numerical experiments comparing FHIRKA to other algorithms further support our theoretical results.
Next, we discuss model reduction for nonlinear dynamical systems. For models with unstructured nonlinearities, POD is the method of choice. However, POD is input dependent and not optimal with respect to the output. Thus, we use operator splitting to integrate the best features of system theoretic approaches with trajectory based methods such as POD in order to mitigate the effect of the control inputs for the approximation of nonlinear dynamical systems. We reduce the linear terms with system theoretic methods and the nonlinear terms terms via POD. Evolving the linear and nonlinear terms separately yields the reduced operator splitting solution. We present an error analysis for this method, as well as numerical results that illustrate the effectiveness of our approach. While in this dissertation we only pursue the splitting of linear and nonlinear terms, this approach can be implemented with Quadratic Bilinear IRKA or Balanced Truncation for Quadratic Bilinear systems to further diminish the input dependence of the reduced order modeling. / Doctor of Philosophy / Simulation, design, and control of dynamical systems play an important role in numerous scientific and industrial tasks such as signal propagation in the nervous system, heat dissipation, electrical circuits and semiconductor devices, synthesis of interconnects, prediction of major weather events, spread of fires, fluid dynamics, machine learning, and many other applications. The need for detailed models leads to large-scale dynamical systems, posing tremendous computational difficulties when applied in numerical simulations. In order to overcome these challenges, we perform model reduction, replacing the large-scale dynamics with high-fidelity reduced representations. Reduced order modeling helps us to avoid the outstanding burden on computational resources. Numerous model reduction techniques exist for linear models over an infinite horizon. However, in practice we usually are interested in reducing a model over a specific time interval. In this dissertation, given a reduced order, we present a method that finds the best local approximation of a dynamical system over a finite horizon. We present both theoretical and numerical evidence that supports the proposed method.
We also develop an algorithm that integrates operator splitting with model reduction to solve nonlinear models more efficiently while preserving a high level of accuracy.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/99358 |
| Date | 15 July 2020 |
| Creators | Sinani, Klajdi |
| Contributors | Mathematics, Gugercin, Serkan, Beattie, Christopher A., Borggaard, Jeffrey T., Embree, Mark P. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
Page generated in 0.0022 seconds