A reduced and limited‐memory preconditioned approach for the 4D‐Var data‐assimilation problem

We recall a theoretical analysis of the equivalence between the Kalman filter and the four-dimensional variational (4D-Var) approach to solve data-assimilation problems. This result is then extended to cover the comparison of the singular evolutive extended Kalman (SEEK) filter with a reduced variant of the 4D-Var algorithm. We next concentrate on the solution of the 4D-Var, which is usually computed with a (truncated) Gauss–Newton algorithm using a preconditioned conjugate-gradient-like (CG) method. Motivated by the equivalence of the above-mentioned algorithms, we explore techniques used in the SEEK filter and based on empirical orthogonal functions (EOFs) as an attempt to accelerate the Gauss–Newton method further. This leads to the development of an appropriate starting point for the CG method, together with that of a powerful limited-memory preconditioner (LMP), as shown by preliminary numerical experiments performed on a shallow-water model. Copyright © 2011 Royal Meteorological Society

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