Algorithms for data assimilation try to predict the most likely state of a dynamical system by combining information from observations and prior models. Variational approaches, such as the weak-constraint four-dimensional variational data assimilation formulation considered in this problem, can ultimately be interpreted as a minimization problem. One of the main challenges of such a formulation is the solution of large linear systems of equations which arise within the inner linear step of the adopted nonlinear solver. Depending on the adopted approach, these linear algebraic problems amount to either a saddle point linear system or a symmetric positive definite (SPD) one. Both formulations can be solved by means of a Krylov method, like GMRES or CG, that needs to be preconditioned to ensure fast convergence in terms of the number of iterations. In this paper we illustrate novel, efficient preconditioning operators which involve the solution of certain Stein matrix equations. In addition to achieving better computational performance, the latter machinery allows us to derive tighter bounds for the eigenvalue distribution of the preconditioned linear system for certain problem settings. A panel of diverse numerical results displays the effectiveness of the proposed methodology compared to current state-of-the-art approaches.

翻译：数据同化算法旨在通过融合观测信息与先验模型来预测动力系统的最可能状态。本文所研究的弱约束四维变分数据同化公式等变分方法，本质上可视为一个最小化问题。此类公式的主要挑战之一，源于所采用的非线性求解器内部线性步中产生的大型线性方程组的求解问题。根据所选方法的不同，这些线性代数问题或归结为鞍点线性系统，或归结为对称正定系统。这两种形式均可通过Krylov子空间方法（如GMRES或CG）求解，但为确保迭代次数层面的快速收敛，必须引入预条件算子。本文提出了新型高效预条件算子，其核心涉及特定Stein矩阵方程的求解。除实现更优的计算性能外，该机制还允许针对特定问题设置推导出预条件线性系统特征值分布的更紧致界。多样化的数值结果表明，与当前最优方法相比，所提方法具有显著有效性。