Sparse system identification by low-rank approximation

In this document, some general results from approximation theory and matrix analysis with applications to the approximate sparse identification of time series models and nonlinear discrete-time dynamical systems are presented. The aforementioned theoretical methods are translated into predictive algorithms that can be used to approximately simulate the behavior of a given dynamical system, based on some structured data measured from the system. The approximation of the state-transition operators determined primarily by matrices of parameters to be identified based on data measured from a given system, is approached by proving the existence of low-rank approximations of submatrices of the trajectory matrices corresponding to the measured data, that can be used to compute approximate sparse representations of the matrices of parameters. Prototypical algorithms and numerical implementations of the aforementioned techniques to the approximate identification and predictive simulation of time series models with symmetries and nonlinear structured dynamical systems in theoretical physics, fluid dynamics and weather forecasting are presented.