Daniel Hodyss, Naval Research Laboratory
Abstract for Special Seminar on Wednesday, October 19 at 3:30 pm in Math 101
*Refreshments at 3:00 pm in Math 101*
In geophysical applications we are rarely if ever able to simulate the problem at hand at a resolution for which all important scales of motion are fully resolved. Almost universally we must truncate the continuous variables of interest to a discrete set and then concatenate those variables into a state-vector that does not fully describe the problem. Typically, we model this state-vector with a discretized partial differential equation (PDE) that attempts to model the entire physical system using fewer degrees of freedom than exist in reality. The result of this coarsening of the simulation of the system is that the numerical model does not simulate the actual variables of interest but simulates some (unknown) function of the true variables of interest. For example, a coarse spatial mesh used to solve the typical hyperbolic PDEs of geophysical fluid dynamics delivers a solution that is smoother than a fine spatial mesh, and therefore the solution for each element of the coarse mesh model is some function of many elements of the fine mesh model. Observations of the actual physical system often observe state variables at this higher resolution, which are not actually simulated by our coarse mesh forecast model, at least not directly. How then should we make use of these observations in a data assimilation algorithm? If data assimilation is interpreted as an application of Bayes’ rule then what does Bayes’ rule mean in this context? We will review some recent research aimed at the development of a rigorous Bayesian framework for understanding and addressing these issues.