Multi-index ensemble Kalman filtering

卡尔曼滤波的动态低秩逼近

Dynamical Low-Rank Approximations for Kalman Filtering

提出卡尔曼 - 布西过程的动态低秩逼近（DLR - KBP）及扩展的DLR - ENKF过程，通过在低维时变子空间演化降低计算成本，增大系综规模，降低蒙特卡罗误差，提高滤波精度，经理论研究和数值实验验证有效。

线性不适定问题自适应集成卡尔曼反演的收敛速率研究

On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

本文将集成卡尔曼反演视为Tikhonov正则化的低秩近似，提出基于Nyström方法的采样方案及样本量与正则化参数耦合的自适应版本，并证明其在偏差原则下具有阶数最优性。

Analysis of a localised nonlinear Ensemble Kalman Bucy Filter with complete and accurate observations

Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the Ensemble Kalman Filter with proper localization techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localized Ensemble Kalman Bucy Filter (l-EnKBF) for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long tim…

一种用于集合卡尔曼滤波的局地化技术

A localization technique for ensemble Kalman filters

针对集合卡尔曼滤波中协方差矩阵计算不准确导致的长程虚假相关，提出基于Schur积的局地化技术，通过连续嵌入集合成员的卡尔曼滤波更新为常微分方程（ODE）实现局地化。

Iterative Ensemble Kalman Methods: A Unified Perspective with Some New Variants

This paper provides a unified perspective of iterative ensemble Kalman methods, a family of derivative-free algorithms for parameter reconstruction and other related tasks. We identify, compare and develop three subfamilies of ensemble methods that differ in the objective they seek to minimize and the derivative-based optimization scheme they approximate through the ensemble. Our work emphasizes two principles for the derivation and analysis of iterative ensemble Kalman methods: statistical linearization and continuum limits. Following these guiding principles, we introduce new iterative ensem…

使用扩展卡尔曼滤波在深度神经网络中进行不确定性传播

Uncertainty Propagation in Deep Neural Networks Using Extended Kalman Filtering

本文提出利用扩展卡尔曼滤波在深度神经网络中传播和量化输入不确定性，方法在计算效率上优于现有技术，同时自然地将模型误差纳入输出不确定性。

An extended sampling-ensemble Kalman filter approach for partial data inverse elastic problems

Inverse problems are more challenging when only partial data are available in general. In this paper, we propose a two-step approach combining the extended sampling method and the ensemble Kalman filter to reconstruct an elastic rigid obstacle using partial data. In the first step, the approximate location of the unknown obstacle is obtained by the extended sampling method. In the second step, the ensemble Kalman filter is employed to reconstruct the shape. The location obtained in the first step guides the construction of the initial particles of the ensemble Kalman filter, which is critical …

$l_p$ regularization for ensemble Kalman inversion

Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and the probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to the linear span of the initial ensemble, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the $l_2$ penalty term, preventing overfitting in th…

集合卡尔曼反演中的吉洪诺夫正则化

Tikhonov Regularization Within Ensemble Kalman Inversion

本文提出在集合卡尔曼反演中引入类似吉洪诺夫的Sobolev惩罚以结合先验信息进行正则化，证明了其连续时间极限的集合坍缩性质，并通过数值实验验证了该方法的优势。

局部加权集合卡尔曼方法的若干视角

Perspectives on locally weighted ensemble Kalman methods

本文从函数逼近视角提出局部加权集合卡尔曼反演方法，结合粒子滤波与集合卡尔曼滤波优势，并用数值实验验证其在逼近和反演中的精度。

An Explicit Probabilistic Derivation of Inflation in a Scalar Ensemble Kalman Filter for Finite Step, Finite Ensemble Convergence

This paper uses a probabilistic approach to analyze the converge of an ensemble Kalman filter solution to an exact Kalman filter solution in the simplest possible setting, the scalar case, as it allows us to build upon a rich literature of scalar probability distributions and non-elementary functions. To this end we introduce the bare-bones Scalar Pedagogical Ensemble Kalman Filter (SPEnKF). We show that in the asymptotic case of ensemble size, the expected value of both the analysis mean and variance estimate of the SPEnKF converges to that of the true Kalman filter, and that the variances of…

多级集合卡尔曼滤波

Multilevel ensemble Kalman filtering

本文将多级蒙特卡洛采样策略引入集合卡尔曼滤波的蒙特卡洛步骤中，针对有限维信号演变和噪声离散时间观测，通过多级时间网格实现SDE的多级数值积分，证明多级EnKF在计算成本与近似精度方面优于传统EnKF。

混沌与机器学习动力系统集合卡尔曼滤波器的长时间精度

Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems

本文研究高维动力系统的在线状态估计问题，确立了集合卡尔曼滤波器在混沌系统及机器学习替代动力学模型中的长时间精度，证明了其估计误差在长时间范围内保持较小，验证了机器学习预测模型在集合数据同化中的有效性。

卡尔曼滤波器在随机微分方程中的应用

Application of Kalman Filter in Stochastic Differential Equations

本文探讨将卡尔曼滤波理论（包括扩展卡尔曼滤波和粒子扩展卡尔曼滤波）应用于随机微分方程，通过滤波拟合SDE系统并跟踪其闭式解，以获取更多信息用于基于数据的SDE模型参数。

Ensemble Kalman更新的非渐近分析：有效维度与局部化

Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization

本文对Ensemble Kalman更新进行非渐近分析，证明当先验协方差具有适度有效维度时小集合即可满足需求，并在统一框架下比较多种实现方法，同时提出新的协方差估计界限。

Continuous Limits for Constrained Ensemble Kalman Filter

The Ensemble Kalman Filter method can be used as an iterative particle numerical scheme for state dynamics estimation and control--to--observable identification problems. In applications it may be required to enforce the solution to satisfy equality constraints on the control space. In this work we deal with this problem from a constrained optimization point of view, deriving corresponding optimality conditions. Continuous limits, in time and in the number of particles, allows us to study properties of the method. We illustrate the performance of the method by using test inverse problems from …

Hierarchical Ensemble Kalman Methods with Sparsity-Promoting Generalized Gamma Hyperpriors

This paper introduces a computational framework to incorporate flexible regularization techniques in ensemble Kalman methods for nonlinear inverse problems. The proposed methodology approximates the maximum a posteriori (MAP) estimate of a hierarchical Bayesian model characterized by a conditionally Gaussian prior and generalized gamma hyperpriors. Suitable choices of hyperparameters yield sparsity-promoting regularization. We propose an iterative algorithm for MAP estimation, which alternates between updating the unknown with an ensemble Kalman method and updating the hyperparameters in the r…

带有重采样的集合卡尔曼滤波器

Ensemble Kalman Filters with Resampling

本文提出一种结合额外重采样步骤的集合卡尔曼滤波器，通过打破粒子间依赖性改进理论分析，并在数值示例中表现良好。

带局地化的集合卡尔曼滤波器的时间一致精度

Time-uniform accuracy of ensemble Kalman filters with localization

本文证明确定性平方根集合卡尔曼滤波器在线性高斯系统中具有时间一致精度，集合大小取决于有效秩和不稳定子空间维数；对局地化滤波器，集合大小取决于局部维数且随空间块数对数增长。