Geometric multiscale reduction for autonomous and controlled nonlinear systems

Most generic approaches to empirical reduction of dynamical systems, controlled or otherwise, are global in nature. Yet interesting systems often exhibit multiscale structure in time or in space, suggesting that localized reduction techniques which take advantage of this multiscale structure might provide better approximations with lower complexity. We introduce a snapshot-based framework for localized analysis and reduction of nonlinear systems, based on a systematic multiscale decomposition of the statespace induced by the geometry of empirical trajectories. A given system is approximated by a piecewise collection of low-dimensional systems at different scales, each of which is suited to and responsible for a particular region of the statespace. Within this framework, we describe localized, multiscale variants of the proper orthogonal decomposition (POD) and empirical balanced truncation methods for model order reduction of nonlinear systems. The inherent locality of the treatment further motivates control strategies involving collections of simple, local controllers and raises decentralized control possibilities. We illustrate the localized POD approach in the context of a high-dimensional fluid mechanics problem involving incompressible flow over a bluff body.