Relaxation-based schemes for on-the-fly parameter estimation in dissipative dynamical systems

Abstract This article studies two particular algorithms, a relaxation least squares algorithm and a relaxation Newton iteration scheme, for reconstructing unknown parameters in dissipative dynamical systems. Both algorithms are based on a continuous data assimilation (CDA) algorithm for state reconstruction of Azouani et al (2014 J. Nonlinear Sci. 24 277–304). Due to the CDA origins of these parameter recovery algorithms, these schemes provide on-the-fly reconstruction, that is, as data is collected, of unknown state and parameters simultaneously. It is shown how both algorithms give way to a robust general framework for simultaneous state and parameter estimation. In particular, we develop a general theory, applicable to a large class of dissipative dynamical systems, which identifies structural and algorithmic conditions under which the proposed algorithms achieve reconstruction of the true parameters. The algorithms are implemented on a high-dimensional two-layer Lorenz 96 model, where the theoretical conditions of the general framework are explicitly verifiable. They are also implemented on the two-dimensional Rayleigh–Bénard convection system to demonstrate the applicability of the algorithms beyond the finite-dimensional setting. In each case, systematic numerical experiments are carried out probing the efficacy of the proposed algorithms, in addition to the apparent benefits and drawbacks between them.