Approximate Gauss-Newton methods using reduced order models

The Gauss-Newton (GN) method is a well known iterative technique for solving nonlinear least squares problems subject to dynamical system constraints. Such problems arise commonly from applications in optimal control and state estimation. Variational data assimilation systems for weather, ocean and climate prediction currently use approximate GN methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new method for deriving low order system approximations based on model reduction techniques from control theory. We show how this technique can be combined with the GN method to give a state estimation technique that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard truncation techniques.