Approximate Gauss–Newton methods for optimal state estimation using reduced‐order models

Abstract

SUMMARYThe 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 in optimal state estimation where the systems may be stochastic. Variational data assimilation techniques for state estimation in weather, ocean and climate systems 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 retain the response of the dynamical system more accurately and improve the performance of the approximate GN method.

Numerical linear algebra in data assimilation

“…In [71] it is assumed that the time-dependent system underlying the problem has a time-invariant dominant part on which balanced truncation is performed. MOR via balanced truncation was also proposed for incremental 4D-Var in [25,26,124,125]. Model and observation operators are projected as in (34), x is restricted tox = U T x ∈ R r , r ≪ n, and the background error covariance matrix, B, is projected onto U T BU.…”

Numerical Linear Algebra in Data Assimilation

“…In [66] it is assumed that the time-dependent system underlying the problem has a time-invariant dominant part on which balanced truncation is performed. MOR via balanced truncation was also proposed for incremental 4D-Var in [24,25,119,118]. Model and observation operators are projected as in (32), δx is restricted to δx = U T δx ∈ R r , r ≪ n, and the background error covariance matrix is projected onto U T BU .…”

Variational data assimilation for very large environmental problems

“…In this method the snapshot perturbations X are weighted according the sensitivity of the cost function at the time of the snapshot, where the weights are calculated using the adjoint model [30]. The other approach, put forward in the series of papers [76], [75], [14], is to use near-optimal model order reduction methods for linear dynamical systems to derive a reduced order model and observation operator. The inner loop problem of incremental 4D-Var (2.15) is subject to the dynamical system described by the evolution equation (216) and the output equation…”

“…The reduction step then calculates the restriction and prolongation operators from 35) where the decay of the Hankel singular values is used to choose the model reduction order r. In idealised models the studies [76], [75], [14] show how this method improves the solution with respect to using low resolution models and how it is important to use information about the assimilation problem in the reduction procedure, including information about the background and observation error covariance matrices. However, whereas reduction methods based on POD can be implemented in large systems, the method of balanced truncation cannot.…”