## Abstract

Incremental four-dimensional variational assimilation (4D-Var) is an algorithm that approximately solves a nonlinear minimization problem by solving a sequence of linearized (quadratic) minimization problems of the form min_{x} F [x] = 1/2 x^{T}Ax - b^{T}x + c, where x is the control vector, A is a symmetric positive-definite matrix, b is a vector containing data and prior information, and c is a constant. This paper proposes a family of limited-memory preconditioners (LMPs) for accelerating the convergence of conjugate-gradient (CG) methods used to solve quadratic minimization problems such as those encountered in incremental 4D-Var. The family is constructed from a set of vectors {s_{i}:i = 1,..., l}, where each s_{i} is assumed to be conjugate with respect to the (Hessian) matrix A. In incremental 4D-Var, approximate LMPs from this family can be built using conjugate vectors generated during the CG minimization on previous outer iterations. The spectral and quasi-Newton LMPs employed in many operational 4D-Var systems are shown to be special cases of the family of LMPs proposed here. In addition, a new LMP based on Ritz vectors (approximate eigenvectors) is derived. The Ritz LMP can be interpreted as a stabilized version of the spectral LMP. Numerical experiments performed with a realistic global ocean 4D-Var system are presented, to test the impact of the three different preconditioners. The Ritz LMP is shown to be more effective than, or at least as effective as, the spectral and quasi-Newton LMPs in our 4D-Var experiments. Our experiments also demonstrate the importance of limiting the number of CG (inner) iterations on certain outer iterations to avoid possible divergence of the cost function on the outer loop. The optimal number of CG iterations will depend on the specific preconditioner used, and can be computed a priori, albeit at the expense of several evaluations of the cost function on the outer loop. In a cycled 4D-Var system, it may be necessary to perform this computation periodically to account for changes in the Hessian matrix arising from changes in the observing system and background-flow field.

Original language | English |
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Pages (from-to) | 751-769 |

Number of pages | 19 |

Journal | Quarterly Journal of the Royal Meteorological Society |

Volume | 134 |

Issue number | 632 |

DOIs | |

Publication status | Published - 2008 |

## Keywords

- 4D-Var
- Conjugate gradient
- Minimization
- Preconditioning