A Generic Bilevel Formalism for Unifying and Extending Model Reduction Methods

Abstract

Abstract : An abstract, algebraic bilevel version of conventional multigrid methods has been developed that formally unifies and extends the reduced basis, substructuring, smoothing/homogenization, and frequency window methods of model reduction. Within the context of this formalism, a given model reduction method synthesizes a reduced model that plays the role of a model. Another original-model approximation, conjugate to the model reduction method, plays the role of a fine model. Conventional multigrid methods can be thought of as an extension of the coarse grid model beyond the original scope of its accuracy through the alternating use offine-grid-error-smoothing and coarse-grid-correction steps. Analogously fashioned, abstract versions of these steps extend the particular model reduction method used beyond the original scope of its accuracy. In addition to the development of the mathematical formalism, a generic continuation is proposed and developed as the conjugate approximation for use in the abstract version of the fine-grid-error-smoothing step. For generality, the nature of the parameter-embedding of the continuation is deliberately left unspecified; a particular parameter-embedding is chosen by the analyst to complement the particular model and model reduction method (or class thereof) in a given case. As an example, a particular submethod of this formalism is constructed by specializing the parameter-embedding to a temporal, biscale perturbational parameter-embedding for the case of a generic set of coupled ordinary differential equations with constant coefficients. The initial iteration of this submethod is a general, combined transient/frequency-window methodology for linear finite-element method applications. It is shown to encompass both the force derivative and Lanczos/Ritz vector methods for the limiting case of a zero frequency, single-point window for the fast time scale.