Many modern algorithms for inverse problems and data assimilation rely on ensemble Kalman updates to blend prior predictions with observed data. Ensemble Kalman methods often perform well with a small ensemble size, which is essential in applications where generating each particle is costly. This paper develops a non-asymptotic analysis of ensemble Kalman updates that rigorously explains why a small ensemble size suffices if the prior covariance has moderate effective dimension due to fast spectrum decay or approximate sparsity. We present our theory in a unified framework, comparing several implementations of ensemble Kalman updates that use perturbed observations, square root filtering, and localization. As part of our analysis, we develop new dimension-free covariance estimation bounds for approximately sparse matrices that may be of independent interest.

翻译：许多应用于逆问题与数据同化的现代算法依赖集成卡尔曼更新来融合先验预测与观测数据。集成卡尔曼方法通常能在小集合规模下表现良好，这在每个粒子生成成本高昂的应用中至关重要。本文对集成卡尔曼更新进行了非渐近分析，严格解释了当先验协方差因快速谱衰减或近似稀疏性而具有适中有效维度时，为何小集合规模足以满足要求。我们在统一框架下阐述理论，比较了使用扰动观测、平方根滤波和局部化技术的多种集成卡尔曼更新实现。作为分析的一部分，我们为近似稀疏矩阵建立了新的无维协方差估计界，该结果可能具有独立的研究价值。