Quadratic Inverse Eigenvalue Problem and Its Applications to Model Updating — An Overview

Modeling is one of the most fundamental tools that we use to simulate the complex world. The goal of modeling is to come up with a representation that is simple enough for mathematical manipulation yet powerful enough for describing, inducing, and reasoning complicated phenomena. When modeling physical systems, the resulting mathematical models are sometimes of a very high order too expensive for simulation. One remedy is the notion of model reduction that assists in approximating very high order mathematical models with lower order models. As is evidenced in this collection, model reduction has been under extensive study and rapid development over the past few years with many physical and engineering applications. On the other hand, precise mathematical models of physical systems are hardly available in practice. Many factors, including inevitable disturbances to the measurement and imperfect characterization of the model, attribute to the inexactitude. Since the model reduction process begets only a partial effect of the original model, it is reasonable to expect that the reduced model might not be consonant with realistic data either. For various reasons, it often becomes necessary to update a primitive model to attain consistency with empirical results. This procedure of updating or revising an existing model is another essential ingredient for establishing an effective model. The emphasis of the following discussion is on the model updating of quadratic pencils.