论文标题
多级合奏Kalman-bucy过滤器
Multilevel Ensemble Kalman-Bucy Filters
论文作者
论文摘要
在本文中，我们考虑了连续时间的线性过滤问题。我们为集合Kalman-Bucy过滤器（ENKBFS）开发并应用多级蒙特卡洛（MLMC）策略。这些过滤器可以看作是有条件的麦基恩 - 维拉索夫型扩散过程的近似值。它们也被解释为\ textit {集成kalman滤波器}的连续时间类似物，由于其适用性和计算成本，事实证明，它已成功。我们证明，我们的多级ENKBF的理想版本可以实现$ \ Mathcal {O}（ε^2），\ε> 0 $的均值误差（MSE），并具有订单成本$ \ MATHCAL {O}（ε^{ - 2} \ log log log（ε）^2）^2）$。为了证明这一结果，我们提供了蒙特卡洛收敛性和与时间限制的ENKBF相关的近似界限。这意味着与（单级）ENKBF相比，成本的降低，该成本需要$ \ MATHCAL {O}（ε^{ - 3}）$，以实现$ \ Mathcal {O}（O}（ε^2）$的MSE。我们测试了关于线性问题的理论，我们通过$ \ sim \ Mathcal {o}（10^4）$和$ \ Mathcal {O}（10^5）$的高维示例来激励该理论。
In this article we consider the linear filtering problem in continuous-time. We develop and apply multilevel Monte Carlo (MLMC) strategies for ensemble Kalman-Bucy filters (EnKBFs). These filters can be viewed as approximations of conditional McKean-Vlasov-type diffusion processes. They are also interpreted as the continuous-time analogue of the \textit{ensemble Kalman filter}, which has proven to be successful due to its applicability and computational cost. We prove that an ideal version of our multilevel EnKBF can achieve a mean square error (MSE) of $\mathcal{O}(ε^2), \ ε>0$ with a cost of order $\mathcal{O}(ε^{-2}\log(ε)^2)$. In order to prove this result we provide a Monte Carlo convergence and approximation bounds associated to time-discretized EnKBFs. This implies a reduction in cost compared to the (single level) EnKBF which requires a cost of $\mathcal{O}(ε^{-3})$ to achieve an MSE of $\mathcal{O}(ε^2)$. We test our theory on a linear problem, which we motivate through high-dimensional examples of order $\sim \mathcal{O}(10^4)$ and $\mathcal{O}(10^5)$.
