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.

翻译：数据同化算法结合观测信息与先验模型信息，以获取动力系统最可能的状态。线性化弱约束四维变分同化问题可重新表述为鞍点问题，与原始形式相比，该表述为预条件子提供了更大设计空间。本文设计了可应用于现有预条件子（如块对角型和约束型预条件子）中的新型项。我们的创新预条件方法：（i）融合模型信息，且（ii）针对相关观测误差协方差矩阵进行设计。据我们所知，（i）此前尚未被考虑用于数据同化问题。我们建立了新理论，论证了新型预条件子在Krylov子空间方法中的有效性。线性和非线性数值实验表明，与现有最优预条件子相比，我们的新方法在更广泛的问题范围内实现了更快收敛——其适用范围超越了纯理论分析所指示的范畴。我们展示了一系列串行执行的数值实验结果。