Nonlinear Bayesian Update via Ensemble Kernel Regression with Clustering and Subsampling
Yoonsang Lee
TL;DR
The paper tackles the limitation of ensemble Kalman filters under non-Gaussian priors and nonlinear measurements by introducing a nonlinear Bayesian update that separates the observed and unobserved components. It uses kernel density estimation to perform nonlinear regression of the unobserved state $u$ on the observed state $v$, evaluating the conditional mean $\mu_{u|v}(v)$ at $\hat{v}$ obtained from a Kalman denoising of the measurements, and enhances robustness with subsampling via the Mahalanobis distance and optional unsupervised clustering. The method provides a practical, nonparametric alternative to Gaussian mixtures or particle filters, with a simple posterior covariance treatment for $u$ and a complete algorithm that switches to the linear update when local data are insufficient. Across Lorenz-63, a PDE-constrained Darcy problem, and high-dimensional Lorenz-96 tests, the nonlinear update reduces estimation errors in strongly nonlinear regimes, particularly when subsampling and clustering are combined, highlighting its potential as a robust augmentation to ensemble-based inference in challenging settings.
Abstract
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.