Ergodic theory in data assimilation
Data assimilation describes the method of blending dynamical models and observational data, with the objective of reducing uncertainty in state estimation and prediction. The procedure has an ’optimal’ Bayesian solution, which tends to be computationally intractable for high dimensional models. As a consequence, many approximation procedures, called approximate filters, have been developed in the geoscience and numerical weather prediction communities, where models tend to be very high dimensional, and where state estimation and uncertainty quantification are central tenets. It is important that we judge these approximations on how well they inherit important features from the true Bayesian solution. In this talk, we will investigate ergodicity for several types of filters that are prevalent in numerical weather prediction. Ergodicity is of crucial importance for filters; it implies a robustness with respect to initial perturbations in state approximations, moreover it suggests that the filter, which is a proxy for the true underlying dynamical system, is inheriting important physical statistical properties. Alongside mostly positive results, we will see that approximate filters don’t always do a good job of inheriting ergodicity. We define a class of models, which are highly stable (and certainly ergodic) for which well trusted approximate filters exhibit strong sensitivity to initialization. In other words, the filters quickly lose touch with reality. This talk is based on several joint works with Andy Majda, Andrew Stuart, Xin Tong, Eric Vanden-Eijnden and Jonathan Weare.