Sigest

Abstract

In applied mathematics and scientific computation, we commonly generate mathematical models of great size and complexity. These can consist of hundreds of thousands or millions of parameters and algebraic or differential equations. Our field consistently makes great advances in the ability to solve larger and larger models both through improved algorithms and effective utilization of ever faster computers.Still, in many situations, it is very important to be able to produce simplified mathematical models that represent the problem well. The reasons for using simplified models range from the efficiency of solving them to their ability to focus upon the primary characteristics of the problem. An overall approach of producing such simplified models is called reduced‐order models, and one of the key techniques for generating reduced‐order models is proper orthogonal decomposition (POD). POD is applied in a variety of fields and applications under a variety of names. For example, in statistics, principal component analysis techniques are used to transform a large set of possibly correlated variables into a smaller set of uncorrelated variables. The method of empirical orthogonal functions performs a related transformation in the area of models of geophysical fluid dynamics.Naturally, a key question is how well reduced‐order models reflect the original mathematical model. A variety of existing research has addressed this question. In this issue’s SIGEST paper, “Error Estimation for Reduced‐Order Models of Dynamical Systems,” by C. Homescu, L. Petzold, and R. Serban, which originally appeared in the SIAM Journal on Numerical Analysis in 2005, the authors take this question one step further. They pose, and address, the question of how well the results of the reduced‐order model generated by POD reflect the solution to perturbations of the original model. These perturbations could come from changes to the initial conditions, or to the model parameters.Based on a combination of the small‐sample statistical condition estimation method and error estimation using the adjoint method, the authors are able to establish ranges of perturbations to the original model for which the estimates produced by the reduced‐order model retain a reasonable degree of validity. Their approach produces a type of condition number that can be calculated a priori and that indicates the sensitivity of the results of the reduced‐order model to changes in the parameters of the original model. Their mathematical analysis is followed by several numerical examples that validate their mathematical estimates.We hope that this paper will help introduce SIAM readers to the important area of reduced‐order models, as well as to important new research about the applicability of these models.