Data Assimilation Seminar
Prof. Ming-Cheng Shiue (May 7, 2019, 15:30-)
|Affiliation||National Chiao Tung University|
|Title||Mathematical analysis of data assimilation algorithms based on synchronization of truth and models|
|Abstract||In this talk, we first recall continuous and discrete data assimilation algorithms that were proposed for designing finite-dimensional feedback controls for two-dimensional Navier-Stokes equations. Then, two new nudging methods, hybrid nonlinear and delay-coordinate nudging are considered and studied.
In the first part, hybrid nonlinear continuous data assimilation algorithms for Lorenz systems will be studied and presented. In the literature, Pecora and Carrol (1990) considered a linear synchronization for a three-variable Lorenz model and numerically found that observing the z variable did not lead to synchronization while observing x or y did. Later, for the same type of synchronization, it was found that synchronizing with y observations is more efficient than with x in Yang et al. (2006). These phenomena will be proven mathematically and explained. Furthermore, three-type hybrid nonlinear nudging techniques are considered to speed up the convergence of rate for the linear nudging one. It is shown that the approximate solutions converge to the unknown reference solutions over time provided that the first or second variable of Lorenz systems is only synchronized. Numerical simulations are performed to demonstrate these results. This is joint work with Yi Juna Du.
In the second part, two new continuous and discrete data assimilation algorithms for two-dimensional Navier-Stokes equations are presented and studied. The explicit use of present and past observations at each time step provides a way that new methods might outperform the old one, which was successfully tested for Lorenz 96 model. In this talk, we will give preliminary results that provide sufficient conditions on the finite-dimensional spatial resolution of the collected data and observational measurements to make sure that the approximate solutions obtained from the new algorithms converge to the unknown reference solutions over time.