Non-asymptotic analysis of ensemble Kalman updates: effective dimension and localization
Non-asymptotic analysis of ensemble Kalman updates: effective dimension and localization
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DOI：
10.1093/imaiai/iaad043
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发表时间：
2024-01-01
影响因子：
1.6
通讯作者：
Sanz-Alonso，Daniel
中科院分区：
文献类型：
--
作者：
Al-Ghattas，Omar;Sanz-Alonso，Daniel
许多反问题和数据同化的现代算法依赖于集合卡尔曼更新，以融合先前的预测与观测数据。Ensemble Kalman方法通常在小的集合尺寸下表现良好，这在生成每个粒子的成本很高的应用中是必不可少的。本文发展了一个非渐近分析的集合卡尔曼更新，严格地解释了为什么一个小的集合大小足够，如果先验协方差具有适度的有效尺寸，由于快速谱衰减或近似稀疏。我们提出了我们的理论在一个统一的框架，比较几种实现的集合卡尔曼更新，使用扰动观测，平方根滤波和本地化。作为我们分析的一部分，我们开发了新的无量纲协方差估计界近似稀疏矩阵，可能是独立的利益。
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, which 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.