Fundamental law preservation in model reduction

Abstract

A preceding paper [1] provides an abstract setting for the process of mathematically reducing, or simplifying, a complicated mathematical model of a physical system to a more manageable, intelligible model. A model reduction framework is developed in this paper which modifies the formalism of the preceding paper so as to allow, in the model synthesis process, the direct preservation of part of the original, complex model in the reduced model. This revised formalism encompasses the previous formalism as a special case. The revised formalism can be used to guide the development of more specific model reduction methods for those cases for which the preservation, in the reduced model, of the appropriate fundamental laws found in the original model is vital. As a common example, any reduced model in a continuum physics setting should include the universal, material-independent version of the appropriate conservation laws found in the original model if it is to predict responses such that one can still associate a physical interpretation with each response. The revised formalism is given entirely in terms of mapping and the required algebraic properties of these mappings. Since these properties are expressed exclusively in terms of mapping composition, this formalism establishes a very general mathematical foundation for the subject. It is used to outline a possible approach to implementing the model reduction process, based upon trial mappings and norm minimization, for nonlinear problems. The formalism is extended so as to allow a model reduction method developed for systems described in one space (such as physical space), to be utilized for model reduction of systems described in other spaces (such as wave number space).