Unstable dynamical systems can be stabilized, and hence the solution recovered from noisy data, provided two conditions hold. First, observe enough of the system: the unstable modes. Second, weight the observed data sufficiently over the model. In this talk I will illustrate this for the 3DVAR filter applied to three dissipative dynamical systems of increasing dimension: the Lorenz 1963 model, the Lorenz 1996 model, and the 2D Navier-Stokes equation.
Kody Law is a Postdoctoral Research Fellow at the Warwick Mathematics Institute at the University of Warwick, where he has been since completion of his PhD in February, 2010. He completed his PhD at the University of Massachusetts in Amherst. His current research focus is on data assimilation and uncertainty quantification, from both theoretical and practical perspectives. He is also involved in bifurcation and stability analysis of nonlinear PDEs.
His interest in data assimilation relates particularly to the accuracy and stability of filters for high-dimensional systems and the relationship to both nonlinear filtering as a sequential Bayesian inverse problem, on the one hand, and deterministic feedback control, on the other hand. Examples of systems which are currently of particular interest are fluid dynamical systems as related to numerical weather prediction and oceanography and also subsurface systems as related to subsurface reconstruction and oil recovery. He is generally interested in both static and online quantification of uncertainty in high-dimensional systems, from both theoretical and practical perspectives. Other such systems may arise, for example, as the discretization of another physical system governed by a PDE.
His research in bifurcation and stability analysis is primarily applied to nonlinear wave equations and, in particular, nonlinear optical systems and Bose-Einstein condensates.