The prototypical problem of data assimilation is weather forecasting. The goal is to provide state predictions for nonlinear and extremely high dimensional models, the novelty is to determine how best to guide these predictions using the now vast amounts of observational data that is available to forecasters. Most data assimilation algorithms have been developed within the applied sciences, notably in meteorology. As a consequence these algorithms sit on little to no mathematical foundation, instead they are ad hoc methods whereby well-understood algorithms are tweaked to make them more suitable to the given problem. Nevertheless, numerical evidence suggests that these algorithms work - they often possess mathematically well understood properties, such as filter accuracy (tracking the right signal) and stability (ergodicity).
We will discuss two roles that mathematicians can play in this field: 1 - to develop understanding of the mathematics behind seemingly ad hoc forecasting algorithms and 2 - to use this understanding to propose new algorithms that are strongly backed by theory.
This is based on several joint works with Andy Majda, Andrew Stuart and Xin Tong.