A New Look at Proper Orthogonal Decomposition

We investigate some basic properties of the proper orthogonal decomposition (POD) method as it is applied to data compression and model reduction of finite dimensional nonlinear systems. First we provide an analysis of the errors involved in solving a nonlinear ODE initial value problem using a POD reduced order model. Then we study the effects of small perturbations in the ensemble of data from which the POD reduced order model is constructed on the reduced order model. We explain why in some applications this sensitivity is a concern while in others it is not. We also provide an analysis of computational complexity of solving an ODE initial value problem and study the computational savings obtained by using a POD reduced order model. We provide several examples to illustrate our theoretical results.