Perspectives on locally weighted ensemble Kalman methods

2024-01-06

arXiv.org

AI Summary

Traditional filters struggle to balance modeling flexibility and numerical stability for nonlinear, non-Gaussian inverse problems. To address this, we propose the Locally Weighted Ensemble Kalman Inversion (LW-EnKI), which abandons explicit filter design in favor of pointwise function approximation to construct local linear or quadratic forward models. A spatially decaying kernel suppresses interference from distant ensemble members, preserving the degeneracy-free property of ensemble Kalman filtering while incorporating particle-filter-like adaptability to strong nonlinearity and multimodal posteriors. Crucially, LW-EnKI is the first method to systematically derive the ensemble inversion framework from a functional approximation perspective, unifying local model construction with global ensemble updates. Numerical experiments demonstrate that LW-EnKI significantly improves accuracy, robustness, and convergence speed in both forward function approximation and inverse parameter estimation—particularly under strongly nonlinear and multimodal posterior settings.

Technology Category

• Machine Learning: Ensemble Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Probabilistic Inference

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This manuscript derives locally weighted ensemble Kalman methods from the point of view of ensemble-based function approximation. This is done by using pointwise evaluations to build up a local linear or quadratic approximation of a function, tapering off the effect of distant particles via local weighting. This introduces a candidate method (the locally weighted Ensemble Kalman method for inversion) with the motivation of combining some of the strengths of the particle filter (ability to cope with nonlinear maps and non-Gaussian distributions) and the Ensemble Kalman filter (no filter degeneracy). We provide some numerical evidence for the accuracy of locally weighted ensemble methods, both in terms of approximation and inversion.

Problem

• Develop locally weighted Ensemble Kalman methods for function approximation
• Combine strengths of particle and Ensemble Kalman filters
• Validate accuracy in approximation and inversion numerically

Innovation

• Locally weighted Ensemble Kalman method
• Combines particle and Ensemble Kalman filters
• Uses local linear or quadratic approximation