We consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block structure, or further to symmetric positive definite systems. In this article, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.

中文翻译：

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弱约束四维变分数据同化中产生的鞍点矩阵的谱估计

我们考虑在稀疏约束的多维变分数据同化的解决方案中出现的大型稀疏对称线性方程组，这是数值天气预报的一种重要方法。这些系统可以写为具有3×3块结构的鞍点系统，但是可以执行块消除以将它们减少为具有2×2块结构的鞍点系统，或者进一步简化为对称正定系统。在本文中，我们分析了这些矩阵的光谱对基本动力系统的观测值有多敏感。我们还获得了矩阵特征值的界限。数值实验用于确定理论分析和界限。