An Adaptive, Black-Box Model Order Reduction Algorithm Using Radial Basis Functions

This project investigates reduced order modeling, as applied to electromagnetics problems. Given a linear system describing a problem and depending on one or more parameters (such as frequency or material properties), model order reduction seeks to find a small set of global basis functions that can accurately approximate the solution across the given range of parameter values. By solving the system in terms of this smaller basis, it is possible to recast the original matrix equation into one with much lower dimensionality; this smaller system can then be solved much more rapidly. Many algorithms, and refinements thereof, have been developed to accomplish this goal. Most are limited by the fact that they explicitly or implicitly assume that it is exactly known how the linear system of interest depends on the parameters. Some, such as those based on a Padé approximation, as also limited to the case of a single parameter. The goal of this work is to adapt the model order reduction framework so that it is suitable for use with a general solution technique. That is, as long as the problem of interest can be represented as a continuously varying linear system, the algorithm should be able to perform order reduction, without having to know the detailed dependence on the parameters. To accomplish this, the process of generating a reduced order model is described in terms of simple linear algebra operations between matrices and vectors, so that it is applicable to a wide variety of problem formulations. Having ii done this, the problem then becomes how to approximate the parameter dependence, since the exact behavior is unknown. This is where radial basis function interpolation comes in: it allows the reduced order system to be efficiently approximated even in the case of a multidimensional parameter space with scattered interpolation points. Furthermore, in contrast to polynomial interpolation, for example, whose basis dimension increases rapidly with order, it is possible to add radial basis function one-by-one, which allows for efficient adaptive interpolation. To this end, several combinations of sampling methods and error estimators are described and evaluated. After the most likely candidates are identified, several numerical examples are presented showing that efficient, accurate reduced order models can be generated, by the same program, for a variety problems, solver formulations, and parameter dependencies.