Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ensemble Kalman Filters with Resampling

Omar Al Ghattas, Jiajun Bao, Daniel Sanz-Alonso

TL;DR

This work addresses online state estimation in high-dimensional systems by enhancing ensemble Kalman filters with a resampling step (REnKF) to break particle dependencies. The authors prove non-asymptotic, dimension-free error bounds for the mean and covariance estimates under linear dynamics, showing that errors scale as the square root of the effective dimension over the ensemble size, with bounds that hold for any fixed $N$ and degrade gracefully over time without assuming ergodicity. The theoretical results are complemented by numerical experiments on linear and Lorenz-96 models, demonstrating that RE n KF achieves performance comparable to standard EnKF while providing stronger theoretical guarantees and robustness to noise. The findings offer a principled pathway to incorporate resampling into EnKF frameworks, enabling reliable long-time error control in high-dimensional filtering tasks and guiding future work on richer resampling schemes and nonlinear dynamics.

Abstract

Filtering is concerned with online estimation of the state of a dynamical system from partial and noisy observations. In applications where the state of the system is high dimensional, ensemble Kalman filters are often the method of choice. These algorithms rely on an ensemble of interacting particles to sequentially estimate the state as new observations become available. Despite the practical success of ensemble Kalman filters, theoretical understanding is hindered by the intricate dependence structure of the interacting particles. This paper investigates ensemble Kalman filters that incorporate an additional resampling step to break the dependency between particles. The new algorithm is amenable to a theoretical analysis that extends and improves upon those available for filters without resampling, while also performing well in numerical examples.

Ensemble Kalman Filters with Resampling

TL;DR

This work addresses online state estimation in high-dimensional systems by enhancing ensemble Kalman filters with a resampling step (REnKF) to break particle dependencies. The authors prove non-asymptotic, dimension-free error bounds for the mean and covariance estimates under linear dynamics, showing that errors scale as the square root of the effective dimension over the ensemble size, with bounds that hold for any fixed and degrade gracefully over time without assuming ergodicity. The theoretical results are complemented by numerical experiments on linear and Lorenz-96 models, demonstrating that RE n KF achieves performance comparable to standard EnKF while providing stronger theoretical guarantees and robustness to noise. The findings offer a principled pathway to incorporate resampling into EnKF frameworks, enabling reliable long-time error control in high-dimensional filtering tasks and guiding future work on richer resampling schemes and nonlinear dynamics.

Abstract

Filtering is concerned with online estimation of the state of a dynamical system from partial and noisy observations. In applications where the state of the system is high dimensional, ensemble Kalman filters are often the method of choice. These algorithms rely on an ensemble of interacting particles to sequentially estimate the state as new observations become available. Despite the practical success of ensemble Kalman filters, theoretical understanding is hindered by the intricate dependence structure of the interacting particles. This paper investigates ensemble Kalman filters that incorporate an additional resampling step to break the dependency between particles. The new algorithm is amenable to a theoretical analysis that extends and improves upon those available for filters without resampling, while also performing well in numerical examples.
Paper Structure (50 sections, 12 theorems, 79 equations, 5 figures, 5 tables, 3 algorithms)

This paper contains 50 sections, 12 theorems, 79 equations, 5 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Resampling in Filtering Algorithms
  3. Our Contributions
  4. Outline
  5. Notation
  6. Problem Setting and Ensemble Kalman Filters
  7. Ensemble Kalman Filters with Resampling
  8. Main Algorithm
  9. Non-asymptotic Error Bounds
  10. Setting and Preliminaries
  11. Main Result
  12. Numerical Results
  13. Linear Dynamics
  14. Effects of Noise Level and Ensemble Size
  15. Effects of State Dimension and Spectrum Decay
  16. ...and 35 more sections

Key Result

Theorem 3.2

\newlabelthm:MultiStepPOEnKFBounds0 Consider $\mathsf{REnKF},$ Algorithm algEnKFresample, with linear dynamics $\Psi(\cdot) = A\cdot$. Suppose that $N \ge r_2(\Sigma^{(0)}) \lor r_2(\Gamma) \lor r_2(\Xi)$. For any $j= 1,2,\dots$, and $q \ge 1$ where $\mu^{(j)}$ and $\Sigma^{(j)}$ are the mean and covariance of the filtering distributions, and $c_1, c_2$ are potentially different universal constan

Figures (5)

  • Figure 1: State estimation and uncertainty quantification for coordinate $u(1)$ in the linear setting with ensemble size $N = 10$ and small noise $\alpha = 10^{-4}.$ Note that the Kaman Filter (KF) is optimal in the linear setting.
  • Figure 2: Effects of $\alpha$ and $N$ in the linear setting with $d=20$.
  • Figure 3: Effect of spectrum decay in the linear setting.
  • Figure 4: State estimation of coordinates $u(1)$ (observed) and $u(3)$ (unobserved) in a partially observed Lorenz 96 system with ensemble size $N = 21$ and small noise $\alpha = 10^{-4}.$$\mathsf{REnKF}$ accurately recovers observed and unobserved coordinates of the state.
  • Figure 5: Effects of $\alpha$, $N,$ and $d$ in the Lorenz 96 example.

Theorems & Definitions (19)

  • Remark 3.1: Deterministic Implementations
  • Theorem 3.2
  • Remark 3.3: Resampled Square-Root Filter
  • Lemma 5.1: Operator Norm of Covariance
  • Proof 1
  • Lemma 5.2: Trace of Offset
  • Proof 2
  • Lemma B.1: Properties of the Kalman Gain Operator kwiatkowski2015convergence
  • Lemma B.2: Properties of the Mean-Update Operator kwiatkowski2015convergence
  • Lemma B.3: Properties of the Covariance-Update Operator kwiatkowski2015convergence
  • ...and 9 more