Sparse system identification by low-rank approximation

In this document, some general results in approximation theory and matrix analysis with applications to sparse identification of time series models and nonlinear discrete-time dynamical systems are presented. The aforementioned theoretical methods are translated into algorithms that can be used for sparse model identification of discrete-time dynamical systems, based on structured data measured from the systems. The approximation of the state-transition operators that are determined primarily by matrices of parameters to be identified based on data measured from a given system, is approached by identifying conditions for 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. One of the main advantages of the low-rank approximation approach presented in this document, concerns the parameter estimation for linear and nonlinear models where numerical or measurement noise could affect the estimates significantly. Prototypical algorithms based on the aforementioned techniques together with some applications to 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.