Leveraging Sparsity and Compressive Sensing for Reduced Order Modeling

Abstract

Sparsity can be leveraged with dimensionality-reduction techniques to characterize and model parametrized nonlinear dynamical systems. Sparsity is used for both sparse representation, via proper orthogonal decomposition (POD) modes in different dynamical regimes, and by compressive sensing, which provides the mathematical architecture for robust classification of POD subspaces. The method relies on constructing POD libraries in order to characterize the dominant, low-rank coherent structures. Using a greedy sampling algorithm, such as gappy POD and one of its many variants, an accurate Galerkin-POD projection approximating the nonlinear terms from a sparse number of grid points can be constructed. The selected grid points for sampling, if chosen well, can be shown to be effective sensing/measurement locations for classifying the underlying dynamics and reconstruction of the nonlinear dynamical system. The use of sparse sampling for interpolating nonlinearities and classification of appropriate POD modes facilitates a family of local reduced-order models for each physical regime, rather than a higher-order global model. We demonstrate the sparse sampling and classification method on the canonical problem of flow around a cylinder. The method allows for a robust mathematical framework for robustly selecting POD modes from a library, accurately constructing the full state space, and generating a Galerkin-POD projection for simulating the nonlinear dynamical system.