Interpolatory Projection Methods for Parameterized Model Reduction

Abstract

Abstract. We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order model. Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reduced-order model. We provide a systematic approach built on established interpolatory H 2 optimal model reduction methods that will produce parameterized reduced-order models having high fidelity throughout a parameter range of interest. For single input/single output systems with parameters in the input/output maps, we provide reduced-order models that are optimal with respect to an H 2 ⊗ L 2 joint error measure. The capabilities of these approaches are illustrated by several numerical examples from technical applications. 1. Introduction. Numerical simulation has steadily increased in importance across virtually all scientific and engineering disciplines. In many application areas, experiments have been largely replaced by numerical simulation in order to save costs in design and development. High accuracy simulation requires high fidelity mathematical models which in turn induce dynamical systems of very large dimension. The ensuing demands on computational resources can be overwhelming and efficient model utilization becomes a necessity. It often is both possible and prudent to produce a lower dimension model that approximates the response of the original one to high accuracy. There are many model reduction strategies in use that are remarkably effective in the creation of compact, efficient, and high fidelity dynamical system models. Such a reduced model can then be used reliably as an efficient surrogate to the original system, replacing it as a component in larger simulations, for example, or in allied contexts that involve design optimization or the development of low-order, fast controllers suitable for real time applications.Typically, a reduced-order model will represent a specific instance of the physical system under study and as a consequence will have high fidelity only for small variations around that base system instance. Significant modifications to the physical model such as geometric variations, changes in material properties, or alterations in

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

“…In accordance with the historical development, we first present the idea of multivariate Padé approximation, or "multimoment matching," and then discuss the more general tangential interpolation approach. The tangential interpolation setting proposed in [29] provides a unifying framework for interpolatory projection-based model reduction of parametric systems and also paves the way to produce optimal (at least locally optimal) parametric reduced models for the composite H 2 ⊗ L 2 error measure.…”

“…A combination of domain decomposition and model reduction has also been developed for optimal control and shape optimization problems [12,13]. Another recent framework employed parametric reduced models that ensure exact matching of the objective function and gradient evaluations for a subset of parameter values [29,80]. Model reduction for optimization problems constrained by partial differential equations (PDEs) has recently been surveyed in [49].…”

“…Even though these types of error measures, which are pointwise in the parameter p, prove effective in practice, the ultimate goal is to minimize a joint error measure defined on a composite frequency and parameter space. The work [29] introduced such a composite error measure, L 2 error in the parameter space and H 2 error in the frequency domain, leading to…”

“…Then, for a specific parameter dependency, [29] introduced a method to minimize this error measure, as explained in section 3.4. One can similarly define a composite error measure in uniform norm in both frequency and parameter domains, namely,…”

“…The ultimate goal is to construct parametric reduced models minimizing these composite measures; however, except for special cases as considered in [29], this goal remains an open challenge.…”