Spatiotemporal system identification with spectral methods - Khanmohamadi, Omid (2009)
In inverse problem theory, the identification of nonlinear spatiotemporal systems is still an underdeveloped topic. This work aims to introduce a means for spatiotemporal system identification based on spectral methods. To achieve this goal, a continuous black-box model class is proposed and parameterized to obtain a model structure whose proper discretization yields a regression form, giving the unknown parameters based on the maximum likelihood estimation. An orthogonal system simplification algorithm is devised to eliminate the redundant parameters. Spectral differentiation operators are extended to inverse problems for proper discretization of the continuous model structure. Trigonometric and algebraic spectral operators are introduced for periodic and non-periodic system identifications and for their practical implementation, wavenumber reordering and roundoff attenuation methods are proposed. The main contribution of this work is to introduce a methodology, namely spectral spatiotemporal system identification, for one-dimensional periodic/non-periodic inverse problems and analyze its superior accuracy and robustness compared with its finite difference counterparts, through the numerical analysis of the method and experiments on indentifying Kuramoto-Sivashinsky (in chaotic regime), Burgers and Allen-Cahn spatiotemporal systems.