An observation‐space formulation of variational assimilation using a restricted preconditioned conjugate gradient algorithm

We consider parameter estimation problems involving a set of m physical observations, where an unknown vector of n parameters is defined as the solution of a nonlinear least-squares problem. We assume that the problem is regularized by a quadratic penalty term. When solution techniques based on successive linearization are considered, as in the incremental four-dimensional variational (4D-Var) techniques for data assimilation, a sequence of linear systems with particular structure has to be solved. We exhibit a subspace of dimension m that contains the solution of these linear systems, and derive a variant of the conjugate gradient algorithm that is more efficient in terms of memory and computational costs than its standard form, when m is smaller than n. The new algorithm, which we call the Restricted Preconditioned Conjugate Gradient (RPCG), can be viewed as an alternative to the so-called Physical-space Statistical Analysis System (PSAS) algorithm, which is another approach to solve the linear problem. In addition, we show that the non-monotone and somehow chaotic behaviour of PSAS algorithm when viewed in the model space, experimentally reported by some authors, can be fully suppressed in RPCG. Moreover, since preconditioning and re-orthogonalization of residuals vectors are often used in practice to accelerate convergence in high-dimension data assimilation, we show how to reformulate these techniques within subspaces of dimension m in RPCG. Numerical experiments are reported, on an idealized data assimilation system based on the heat equation, that clearly show the effectiveness of our algorithm for large-scale problems. Copyright © 2009 Royal Meteorological Society