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

Abstract

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.

Sparse Sensor Placement Optimization for Classification

“…Many applications that require high-performance computing to model dynamical processes in physical systems rely on reduced-order models (ROMs) [48,3] and sensor refinement to streamline computations. As a precursor to modern sparse sensing theory, Everson and Sirovich [21] introduced a method for computing the proper orthogonal decomposition (POD) [31] using incomplete or gappy measurement data.…”

Parametric reduced order model approach for rapid dynamic loads prediction

“…As the generation of a new ROM at each point of interest in the parameter space is usually impractical, and could even be more computationally expensive than building and evaluating the Full Order Model (FOM) anew, Parametric Model Order Reduction (PMOR) has been introduced to efficiently generate ROMs that preserve the parametric dependency and are accurate over a broad range of parameters, without the need of performing a new reduction at each design point. A survey of the state-of-the-art in PMOR is given in [30], where various methodologies are presented and compared.…”

“…As the Balanced Truncation is not a physics-based reduction, the states of the ROMs at different parameter values lie in unrelated subspaces and, before the interpolation, must be transformed, through a similarity transformation 8 ,5 = 5 8 ,¡ ¢ , to a congruent common subspace, spanned by the column of the matrix B ∈ ℝ › • x oe . The choice of this reference subspace is critical for the accuracy of the entire procedure; it is problem-dependent and various options have been proposed [30], [34]. A solution that, for the application considered, is robust and delivers accurate results is adopting the first !…”

Projection-Based Model Reduction for Finite Element Approximation of Shallow Water Flows

“…Thus some form of projection is required to reduce the number of the resulting equations, i.e., multiplication from the left by a full rank matrix of size m ⇥ N . We consider two alternative approaches, namely a standard Galerkin (POD) projection as well as a PetrovGalerkin projection [6,12].…”