Large-scale variational data assimilation problems are commonly found in applications like numerical weather prediction and oceanographic modeling. The 4D-Var method is frequently used to calculate a forecast model trajectory that best fits the available observations to within the observational error over a period of time. One key challenge is that the state vectors used in realistic applications could contain billions or trillions of unknowns so, due to memory limitations, in practice it is often impossible to assemble, store or manipulate the matrices involved explicitly. In this talk we present a limited memory approximation to the Hessian of the linearized quadratic minimization subproblems, computed using the Lanczos method, based on a multilevel approach. We then use this approximation as a preconditioner within 4D-Var and show that it can reduce memory requirements and increase computational efficiency.

Dr. Alison Ramage obtained her PhD from the University of Bristol in 1990 and, since 1993, she has worked at the University of Strathclyde in Glasgow, Scotland, where she is currently a Reader in Industrial and Computational Mathematics. Dr. Ramage's main expertise lies in numerical linear algebra, with a particular focus on developing efficient solution methods for numerical models of partial differential equations and a primary interest in preconditioning linear solvers. Her work has always been highly motivated by practical applications areas such as computational fluid dynamics, soil engineering, option pricing in financial mathematics, modeling liquid crystal displays and, most recently, weather forecasting and environmental modeling.