Projection-Based Model Reduction Using Asymptotic Basis Functions

Galerkin projection provides a formal means to project a differential equation onto a set of preselected basis functions. This may be done for the purpose of formulating a numerical method, as in the case of spectral methods, or formulation of a reduced-order model (ROM) for a complex system. Here, a new method is proposed in which the basis functions used in the projection process are determined from an asymptotic (perturbation) analysis. These asymptotic basis functions (ABF) are obtained from the governing equation itself; therefore, they contain physical information about the system and its dependence on parameters contained within the mathematical formulation. This is referred to as reduced-physics modeling (RPM) as the basis functions are obtained from a physical model-driven, rather than data-driven, technique. This new approach is tailor-made for modeling multiscale problems as the various scales, whether overlapping or distinct in time or space, are formally accounted for in the ABF. Regular- and singular-perturbation problems are used to illustrate that projection of the governing equations onto the ABF allows for determination of accurate approximate solutions for values of the "small" parameter that are much larger than possible with the asymptotic expansion alone and naturally accommodate multiscale problems in which large gradients occur in adjacent regions of the domain.