Dynamical systems are a commonly used and studied tool for simulation, optimization and design. In many applications such as inverse problem, optimal control, shape optimization and uncertainty quantification, those systems typically depend on a parameter. The need for high fidelity in the modeling stage leads to large-scale parametric dynamical systems. Since these models need to be simulated for a variety of parameter values, the computational burden they incur becomes increasingly difficult. To address these issues, parametric reduced models have encountered increased popularity in recent years.
We are interested in constructing parametric reduced models that represent the full-order system accurately over a range of parameters. First, we define a global joint error mea- sure in the frequency and parameter domain to assess the accuracy of the reduced model. Then, by assuming a rational form for the reduced model with poles both in the frequency and parameter domain, we derive necessary conditions for an optimal parametric reduced model in this joint error measure. Similar to the nonparametric case, Hermite interpolation conditions at the reflected images of the poles characterize the optimal parametric approxi- mant. This result extends the well-known interpolatory H2 optimality conditions by Meier and Luenberger to the parametric case. We also develop a numerical algorithm to construct locally optimal reduced models. The theory and algorithm are data-driven, in the sense that only function evaluations of the parametric transfer function are required, not access to the internal dynamics of the full model.
While this first framework operates on the continuous function level, assuming repeated transfer function evaluations are available, in some cases merely frequency samples might be given without an option to re-evaluate the transfer function at desired points; in other words, the function samples in parameter and frequency are fixed. In this case, we construct a parametric reduced model that minimizes a discretized least-squares error in the finite set of measurements. Towards this goal, we extend Vector Fitting (VF) to the parametric case, solving a global least-squares problem in both frequency and parameter. The output of this approach might lead to a moderate size reduced model. In this case, we perform a post- processing step to reduce the output of the parametric VF approach using H2 optimal model reduction for a special parametrization. The final model inherits the parametric dependence of the intermediate model, but is of smaller order.
A special case of a parameter in a dynamical system is a delay in the model equation, e.g., arising from a feedback loop, reaction time, delayed response and various other physical phenomena. Modeling such a delay comes with several challenges for the mathematical formulation, analysis, and solution. We address the issue of transient behavior for scalar delay equations. Besides the choice of an appropriate measure, we analyze the impact of the coefficients of the delay equation on the finite time growth, which can be arbitrary large purely by the influence of the delay. / Ph. D. / Mathematical models play an increasingly important role in the sciences for experimental design, optimization and control. These high fidelity models are often computationally expensive and may require large resources, especially for repeated evaluation. Parametric model reduction offers a remedy by constructing models that are accurate over a range of parameters, and yet are much cheaper to evaluate. An appropriate choice of quality measure and form of the reduced model enable us to characterize these high quality reduced models. Our first contribution is a characterization of optimal parametric reduced models and an efficient implementation to construct them.
While this first framework assumes we have access to repeated evaluations of the full model, in some cases merely measurement data might be available. In this case, we construct a parametric model that fits the measurements in a least squares sense. The output of this approach might lead to a moderate size reduced model, which we address with a post-processing step that reduces the model size while maintaining important properties.
A special case of a parameter is a delay in the model equation, e.g., arising from a feedback loop, reaction time, delayed response and various other physical phenomena. While asymptotically stable solutions eventually vanish, they might grow large before asymptotic behavior takes over; this leads to the notion of transient behavior, which is our main focus for a simple class of delay equations. Besides the choice of an appropriate measure, we analyze the impact of the structure of the delay equation on the transient growth, which can be arbitrary large purely by the influence of the delay.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/83840 |
| Date | 02 July 2018 |
| Creators | Grimm, Alexander Rudolf |
| Contributors | Mathematics, Gugercin, Serkan, Chung, Matthias, Drmac, Zlatko, Embree, Mark P., Beattie, Christopher A., de Sturler, Eric |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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