Nonlinear Bayesian update for a prior ensemble is proposed to extend traditional ensemble Kalman filtering to settings characterized by non-Gaussian priors and nonlinear measurement operators. In this framework, the observed component is first denoised via a standard Kalman update, while the unobserved component is estimated using a nonlinear regression approach based on kernel density estimation. The method incorporates a subsampling strategy to ensure stability and, when necessary, employs unsupervised clustering to refine the conditional estimate. Numerical experiments on Lorenz systems and a PDE-constrained inverse problem illustrate that the proposed nonlinear update can reduce estimation errors compared to standard linear updates, especially in highly nonlinear scenarios.

翻译：本文提出了一种针对先验集合的非线性贝叶斯更新方法，旨在将传统集成卡尔曼滤波推广至非高斯先验与非线性测量算子的场景。在此框架中，观测分量首先通过标准卡尔曼更新进行去噪，而未观测分量则采用基于核密度估计的非线性回归方法进行估计。该方法引入子采样策略以确保稳定性，并在必要时采用无监督聚类技术优化条件估计。通过对Lorenz系统及偏微分方程约束反问题的数值实验表明，相较于标准线性更新方法，所提出的非线性更新策略能够有效降低估计误差，尤其在强非线性场景中表现更为显著。