Optimisation and numerical linear algebra in data assimilation

Data assimilation is a method that combines observations (e.g. real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretised partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we derive the most important variational and sequential data assimilation approaches, in particular three-dimensional and four-dimensional variational data assimilation (3D-Var and 4D-Var), and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation. The data assimilation problem usually results in very large optimisation problems and/or very large linear systems to solve. Hence, the final part of this talk aims to review advances and challenges, in particular from the point of view of numerical linear algebra, within the various data assimilation approaches.