Mixture-Weighted Ensemble Kalman Filter with Quasi-Monte Carlo Transport

TL;DR

This paper introduces a novel mixture-weighted ensemble Kalman filter with importance sampling and quasi-Monte Carlo techniques, improving accuracy and reducing sampling error in high-dimensional Bayesian filtering tasks.

Contribution

It develops a theoretical framework for mixture importance sampling, interprets the EnKF as sampling from Gaussian mixtures, and integrates transported quasi-Monte Carlo for enhanced filtering accuracy.

Findings

Improved filtering accuracy over BPF, EnKF, and standard weighted EnKF.

Elimination of EnKF error plateau due to analysis-target mismatch.

Significant reduction in sampling error using TQMC enhancements.

Abstract

The Bootstrap Particle Filter (BPF) and the Ensemble Kalman Filter (EnKF) are two widely used methods for sequential Bayesian filtering: the BPF is asymptotically exact but can suffer from weight degeneracy, while the EnKF scales well in high dimension yet is exact only in the linear-Gaussian case. We combine these approaches by retaining the EnKF transport step and adding a principled importance-sampling correction. Our first contribution is a general importance-sampling theory for mixture targets and proposals, including variance comparisons between individual- and mixture-based estimators. We then interpret the stochastic EnKF analysis as sampling from explicit Gaussian-mixture proposals obtained by conditioning on the current or previous ensemble, which leads to six self-normalized IS-EnKF schemes. We embed these updates into a broader class of ensemble-based filters and prove consistency and error bounds, including weight-variance comparisons and sufficient conditions ensuring finite-variance importance weights. As a second contribution, we construct transported quasi-Monte Carlo (TQMC) point sets for the Gaussian-mixture laws arising in prediction and analysis, yielding TQMC-enhanced variants that can substantially reduce sampling error without changing the filtering pipeline. Numerical experiments on benchmark models compare the proposed mixture-weighted and TQMC-enhanced filters, showing improved filtering accuracy relative to BPF, EnKF, and the standard weighted EnKF, and that the weighted schemes eliminate the EnKF error plateau often caused by analysis-target mismatch.