A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

TLDR

Large‑scale dynamical system simulations are computationally demanding, and while model reduction has matured for linear, non‑parametric systems, parametric model reduction—addressing systems whose equations depend on parameters such as PDEs and ODEs—has only recently emerged as a vibrant research area. This paper surveys recent contributions to parametric model reduction, aiming to provide a comprehensive resource that highlights low‑cost, accurate modeling across varying parameters. The authors review projection‑based parametric model‑reduction techniques, outlining how each class handles parameter variation and comparing their strengths and weaknesses. Parametric model reduction is crucial for design, control, optimization, and uncertainty quantification, where repeated evaluations across parameter values are needed.

Abstract

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification---settings that require repeated model evaluations over different parameter values.