Model Reduction with Rational Krylov Methods

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Simulation and computing have become a standard required task for the modeling and control of many complex phenomena that are of interest in science and industry. Examples abound and range from acoustic wave propagation and noise suppression in large high-speed vehicles to molecular dynamics and protein folding in rational drug design. The need for greater accuracy leads to inclusion of greater detail in the computer model, with potential coupling to other complex computer models that may require additional simulations that are themselves difficult and expensive. The resulting computational burden can be overwhelming and can create unmanageably large demands on resources. Efficient utilization of the computational model becomes a necessary component of simulations in such large-scale settings. This is the main motivation for model reduction. Often, the original system model behaves very nearly as if it were a simpler system -- but unfortunately not one that is explicitly known beforehand. The goal of model reduction is to extract such a simpler system while mimicking the original full system behavior as closely as possible. The new simpler system can then be used as an efficient surrogate for the original system. The research supported here focuses on Krylov-based projection methods to accomplish this task. These methods have emerged as promising candidates for model reduction in large-scale settings over the last ten years, yet their use still requires improvised elements that are not yet well understood. We believe that our methods will permit a systematic refinement of these ideas and lead to the efficient construction of high-fidelity, in some cases optimal, reduced-order models for large-scale systems with precise estimates of the level of model fidelity that has been maintained. Tools for the analysis, approximation, and control of large-scale, complex system models will be produced as well that are anticipated to contribute to scientific research infrastructure.