论文标题
集合Kalman更新的非反应分析：有效的维度和定位
Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization
论文作者
论文摘要
许多用于反问题和数据同化的现代算法都依赖于集成的Kalman更新，以将先前的预测与观察到的数据融为一体。集合Kalman方法通常在较小的集合尺寸的情况下表现良好，这在生成每个粒子成本高昂的应用中至关重要。本文对集合Kalman更新进行了非反应分析，这严格解释了为什么由于快速衰减或近似稀疏性而导致先前的协方差有效的有效维度，那么小型合奏的大小就足够了。我们在统一的框架中介绍了我们的理论，比较了使用扰动观测值，平方根滤波和本地化的集合卡尔曼更新的几个实现。作为分析的一部分，我们为可能具有独立关注的大约稀疏矩阵开发了新的无维度协方差估计界限。
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.
