Multi-scale Dynamics and Information: Data Driven and Random Dynamical Systems
Friday, June 8, 2018 - 9:30am - 10:20am
First part of this lecture provides theoretical results and numerical demonstration for nonlinear filtering of systems with multiple timescales. This work provides the necessary theoretical bedrock upon which computationally efficient algorithms may be further developed to handle the problem of data assimilation in ever-increasingly higher dimensional complex systems; specifically with a focus on Dynamic Data-Driven Application Systems. A motivating problem for the multiscale correlated filtering problem stems from atmospheric and climatology problems, for example coupled atmosphere-ocean models, which immediately provide a multiscale model with fast atmospheric and slow ocean dynamics. A stochastic version of the Lorenz-96 model is used here -- the full multi-scale system is defined as the truth, whereas a truncated, homogenized version is used as testbed for the data assimilation schemes. Second part of the lecture deals with noisy perturbations of linear delay differential equations (DDE) that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. DDE arise in a variety of areas such as manufacturing systems, biological systems, and control systems. We will present some results of eye-pupil response to incident light.