We design and analyse the performance of a multilevel ensemble Kalman filter method (MLEnKF) for filtering settings where the underlying state-space model is an infinite-dimensional spatio-temporal process. We consider underlying models that needs to be simulated by numerical methods, with discretization in both space and time. The multilevel Monte Carlo sampling strategy, achieving variance reduction through pairwise coupling of ensemble particles on neighboring resolutions, is used in the sample-moment step of MLEnKF to produce an efficent hierarchical filtering method for spatio-temporal models. Under sufficent regularity, MLEnKF is proven to be more efficient for weak approximations than EnKF, asymptotically in the large-ensemble and fine-numerical-resolution limit. Numerical examples support our theoretical findings.

我们设计并分析了多层集合卡尔曼滤波方法（MLEnKF）的性能，该滤波方法用于过滤设置，其中基础状态空间模型是一个无限维的时空过程。我们考虑需要通过数值方法模拟的基础模型，并在时空上离散化。多级蒙特卡洛采样策略通过在邻近分辨率上通过成对粒子的成对耦合实现方差减小，被用于MLEnKF的采样矩步骤中，从而为时空模型产生有效的分层滤波方法。在足够的规律性下，在弱的近似中，MLEnKF被证明比EnKF更有效率，这在大集合和精细数值分辨率的极限下是渐近的。数值例子支持了我们的理论发现。