Efficient 𝐻₂-Based Parametric Model Reduction via Greedy Search
Cooper, Jon Carl
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Dynamical systems are mathematical models of physical phenomena widely used throughout the world today. When a dynamical system is too large to effectively use, we turn to model reduction to obtain a smaller dynamical system that preserves the behavior of the original. In many cases these models depend on one or more parameters other than time, which leads to the field of parametric model reduction. Constructing a parametric reduced-order model (ROM) is not an easy task, and for very large parametric systems it can be difficult to know how well a ROM models the original system, since this usually involves many computations with the full-order system, which is precisely what we want to avoid. Building off of efficient 𝐻-infinity approximations, we develop a greedy algorithm for efficiently modeling large-scale parametric dynamical systems in an 𝐻₂-sense. We demonstrate the effectiveness of this greedy search on a fluid problem, a mechanics problem, and a thermal problem. We also investigate Bayesian optimization for solving the optimization subproblem, and end with extending this algorithm to work with MIMO systems.
General Audience Abstract
In the past century, mathematical modeling and simulation has become the third pillar of scientific discovery and understanding, alongside theory and experimentation. Mathematical models are used every day, and are essential to modern engineering problems. Some of these mathematical models depend on quantities other than just time, parameters such as the viscosity of a fluid or the strength of a spring. These models can sometimes become so large and complicated that it can take a very long time to run simulations with the models. In such a case, we use parametric model reduction to come up with a much smaller and faster model that behaves like the original model. But when these large models vary highly with the parameters, it can also become very expensive to reduce these models accurately. Algorithms already exist for quickly computing reduced-order models (ROMs) with respect to one measure of how "good" the ROM is. In this thesis we develop an algorithm for quickly computing the ROM with respect to a different measure - one that is more closely tied to how the models are simulated.
- Masters Theses