Saddle point preconditioners for weak-constraint 4D-Var

TL;DR

This paper introduces novel preconditioners for weak-constraint 4D-Var data assimilation that incorporate model information and target correlated observation errors, resulting in faster convergence in numerical experiments.

Contribution

The paper develops new preconditioning techniques for saddle point formulations of weak-constraint 4D-Var, incorporating model data and observation error correlations, which were not previously considered.

Findings

Faster convergence of Krylov methods with new preconditioners

Effective for a broader class of problems than existing methods

Validated through extensive numerical experiments

Abstract

Data assimilation algorithms combine information from observations and prior model information to obtain the most likely state of a dynamical system. The linearised weak-constraint four-dimensional variational assimilation problem can be reformulated as a saddle point problem, which admits more scope for preconditioners than the primal form. In this paper we design new terms which can be used within existing preconditioners, such as block diagonal and constraint-type preconditioners. Our novel preconditioning approaches: (i) incorporate model information, and (ii) are designed to target correlated observation error covariance matrices. To our knowledge (i) has not previously been considered for data assimilation problems. We develop new theory demonstrating the effectiveness of the new preconditioners within Krylov subspace methods. Linear and non-linear numerical experiments reveal that our new approach leads to faster convergence than existing state-of-the-art preconditioners for a broader range of problems than indicated by the theory alone. We present a range of numerical experiments performed in serial.